38 solved questions of Trigonometry for Trigonometric Identities | Trigonometric Identities | Trigonometry

Hello pupils. I hope you all are doing good and today we have brought to you the collection of all solved questions of trigonometry related to trigonometric identities in one post. This is a collection of all the solved trigonometric identities questions in this blog.

If sinx-cosx = 0, Find the value of cosecx - cotx.
Answer:

This is just very simple question! What we need to do is find the value of x.
If you see the image above , we get: tanx = 1
And since we have x= tan inverse 1
So, X= 45

Now, to find the value of cosecx -cotx
Put the value of x= 45° and find the answer!

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If sinθ - cosθ = 0, find the value of 'θ'.

Answer:

Explanation:

For solution 1,
We do very simple steps. We first add + cos on both sides and as they are equal! We can then divide the sin by cos putting 1 on the right side. Now we get sin/ cos = tan. And from the value table, we know tan 45° = 1. Therefore θ = 45°

For solution 2,
We multiply both sides by
1. (Sin + Cos)
2. We get (a-b)(a+b) on the LHS which is equal to a^2 - b^2.
3. We know, cos^2 x = 1 - sin^2 x

After this, we use the solving steps and we can get the value of the unknown angle in the question!

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If cos^4 A + cos^2 A =1, then prove that: tan^4 A+ tan^2 A =1.

Answer:

So, to solve this question, first you need to see the image. We have
Cos⁴A + cos²A = 1
When we send cos²A to the right side

Cos⁴A = 1- cos² A

And we know,
Sin²A = 1 - cos²A

So,
Cos⁴A = sin²A
When we divide both by cos²A

We get
Cos²A = tan²A. [ TanA = sinA/cosA ]

When we multiply both sides by 1/ cos²A

We get 1 = tan²A * sec²A [ sec²A = 1/ cos²A]

And, sec²A = tan²A + 1
When we simplify this, we get

1 = tan⁴A + tan²A which is our right hand side.

Therefore, you can solve this Trigonometric Identity, If cos^4 A + cos^2 A =1, then prove that: tan^4 A+ tan^2 A =1
In Trigonometry of Mathematics.

Prove that:

1 / (secA -tanA) - 1/ cosA = 1/ cosA - 1/(secA + tanA)

Answer:

This question is about tricky!
We need to rationalize the first term of LHS initially.
Then change 1/ cosA into secA

Once you rationalize the first term, you will get secA +tanA and -secA which means you will get secA +tanA -secA in the LHS.

Now, don't do a mistake by cancelling the +secA and -secA.

Now, take secA alone which will be equal to your 1/ cosA in the RHS.
Now you need to get - 1/ (secA +tanA)

For this you need to rationalize (tanA - secA) by taking - sign common.

Identity used:

Sec²x - tan²x = 1

This is how you prove the given trigonometric identity,
1 / (secA -tanA) - 1/ cosA = 1/ cosA - 1/(secA + tanA)
Of Trigonometry in mathematics.

Prove that:

(1 + tanΘ)^2 + (1 + cotΘ)^2 = (secΘ + cosecΘ)^2

Answer:

The question is very much easy to solve if you know these basic things:

(a +b)² = a² + 2ab + b²

And

1 + tan²x = sec²x
i.e. sec²x - tan²x = 1

Also

1 + cot²x = cosec²x
i.e. cosec²x - cot²x = 1

And
tanx = sinx / cosx
Cotx = cosx/ sinx

Now, just see the image upside and you can visualize the answer.

Therefore, this is the process how you solve the given trigonometric identity,
(1 + tanΘ)^2 + (1 + cotΘ)^2 = (secΘ + cosecΘ)^2, in Trigonometry of Mathematics.

Prove that:

(cotA + cosecA -1)/ (cotA -cosecA +1) =

(1 +cosA)/ sinA

Answer:

For this question, we will use the Trigonometric identity for value of 1

Cosec² x - cot² x= 1

And above identity is a² -b² which can also be written as (a +b)(a -b)

Remember we will only use the identity for the value 1 in numerator not in the denominator.

So, we will take the terms common in numerator and cancel the equal terms of the numerator and denominator.

Then we will make it equal to the right side.

See the image above to visualize the answer.

And this way you will prove the given trigonometric identity ,
(cotA + cosecA -1)/ (cotA -cosecA +1) =

(1 +cosA)/ sinA,

in Trigonometry.

Prove that: tanα / (1 +tan^2 α)^2 + cotα / (1+ cot^2 α)^2 = sinα. Cosα.

Answer:

Just for our easiness, I have solved and found the value of (1+ cot²α)² first. I opened up the formula of (a+b)² = a² +2ab +b²

And then changed the cot into tan as
Cotx = 1/ tanx

Then I took the LHS and started solving the question .
I took the LCM and performed the basic Mathematics.
Then there is a step where I wrote
cotα * tan⁴α = tan³α
It is because
Cotα = 1/ tanα
So
1/ tanα * tan⁴α = tan³α

Then I also used the Trigonometric identity of 1
Where sec² x - tan² x = 1

And the remaining processes, you can see in the figure above.
This is the way how you solve the given trigonometric identity of Trigonometry in mathematics.

Prove that: (tanΘ +secΘ -1) / (tanΘ -secΘ +1) = (1+ sinΘ)/cosΘ = secΘ + tanΘ

Answer:

Here in this question we have to show that left side is equal to middle side is equal to right side.

For this question, we will use the Trigonometric identity for value of 1

Sec² x - tan² x= 1

And above identity is a² -b² which can also be written as (a +b)(a -b)

Remember we will only use the identity for the value 1 in numerator not in the denominator.

So, we will take the terms common in numerator and cancel the equal terms of the numerator and denominator.

Then we will make it equal to the right side.

And then again we will solve the left side's value and make it equal to the middle side.

See the figure to understand more about this.

Prove that:

tan^2 A/ (tanA -1) + cotA /(1 - tanA) = 1 + secA.cosecA

Answer:

Here, we have the question and we are solving it in terms of tan. So, we will change cotA into 1/tanA

And we will apply the basic Mathematics to solve the question. we will take the LCM and then open up the formula of
a³- b³ = (a-b) (a² +ab +b²)

Trigonometric identity used:
Sec² A - tan²A = 1

Check the above image to visualize the answer.

Prove that:

(3 - 4 sin^2 α) /(1- 3tan^2 α) = (3 -tan^2 α) (4cos^2 α -3)

Answer:

For solving such type of question, we need to understand the concept that we will already have two common terms in LHS and the RHS.

let's find them here. Tan^2 α common so let us operate on it.

Taking LHS
We will not touch the first term until we solve the second term
As you can see in the image. We solved the second term whose numerator will be equal to the second term of RHS.

But since the denominator is not needed in that place we shift it in first term and it will be correct because they are in multiple form.

Then we perform the operations on the first term as well and prove that the LHS is equal to the RHS.

The Trigonometric identity used:

Tan x = sinx /cosx
1/ Cox = secx

This is how you prove that the LHS is equal to the RHS in Trigonometric questions in Trigonometry.

Prove that: (cosA - sinA +1)/ (cosA +sinA -1) = cotA + cosecA

Answer:

To solve these types of question, in most of the cases, you will need to take the minus sign common in both numerator and denominator for easiness.

Then you need to rationalize the denominator.

After that you will need to open up the two formulas. I.e. (a-b)² in the numerator and a²-b² in the denominator.

Then you have to use a Trigonometric identity:

Sin²A + cos²A = 1
Then solve the question as shown in the image.

At last after doing basic Mathematical operations, you will need to remember

1/ sinA = cosecA and cosA/sinA = cotA

This is how you prove the given trigonometric identity in Trigonometry of Mathematics.

Prove that:

tan x/ (secx - 1) - sinx/ (1 + cosx) = 2 cotx

Answer:

Here, we use the simple technique. First we change every terms into sin and cos.
I.e.
Tanx = sinx/ cosx
Secx= 1/ cosx

And then, apply simple mathematics to solve the question as shown in the image above.

You then need to take some terms in common and then cancel the equal terms with opposite signs in the numerator.

Then, you can simply convert the remaining terms into cot. As cotx = cosx/ sinx

So, this is the way how you solve this kind of Trigonometric question and prove the given trigonometric identity in Trigonometry of Mathematics.

Prove that:

(cosA - sinA + 1) / (cosA + sinA +1) = (1 - sinA) / cosA

Answer:

For solving this questions we need to rationalize the denominator. But for easiness, we have taken the sign common in the LHS. Then we have rationalized the denominatora you can see in the image.

Then when we rationalize,we have to open the formula of (a-b)² and a² - b²

Once we do this, we get some we get sin² x + cos² x in numerator which we can write as 1.

Then we have to take the terms common and then cancel the equal terms of the numerator and denominator.

Then we apply simple mathematics to solve the question and prove that the LHS is equal to the RHS.

This way you can solve the given trigonometric identity of Trigonometry in mathematics.

Prove that: (tanΘ + cotΘ)² = (1 + tan²Θ) + (1 + cot²Θ)

Answer:

Before solving this question,
Let us understand

(a+b)² = a² + 2ab + b²

tanΘ * cotΘ = 1

Now solving the question we have,
(TanΘ + cotΘ)² when tanΘ = a and cotΘ = b

We have,
a² + 2ab + b² = tan²Θ + 2 tanΘ cotΘ + cot²Θ

Since tanΘ*cotΘ = 1

We get

Tan²Θ + 2 * 1 + cot²Θ
When we expand 2 as 1 + 1

We get
Tan²Θ +1 + 1+ cot²Θ which is equal to the RHS.

Therefore, you solve this type of trigonometric identity questions in the above-mentioned way in Trigonometry.

Prove that:

(sinΘ - cosΘ + 1) (sinΘ + cosΘ - 1) = secΘ + tanΘ

Answer:

The process I used to solve this question is simple! I rationalized the denominator. But for easiness, to open the formula, I took the sign common in between the numerator and denominator as you can see in the image.

Then when we rationalize,we have to open the formula of (a-b)^2 and a^2 - b^2

Once we do this, we get some we get sin^2 x + cos^2 x in numerator which we can write as 1.

Then we have to take the terms common and then cancel the equal terms of the numerator and denominator.

Then we apply simple mathematics to solve the question and prove that the LHS is equal to the RHS.

This way you can solve the given trigonometric identity of Trigonometry in mathematics.

Prove that:

(1 - sin^4 A) / cos^4 A = 1 + 2 tan^2 A

Answer:

We need to remember the little thing which we used to follow Everytime in Trigonometry. The thing is:

a^2 - b^2 = (a +b) (a -b)

We do the same here and expand the term
1- sin^4 A

Then we use the Trigonometric identity:
1- sin^2 A = cos^2 A

Then we cancel the equal terms from numerator and denominator. I.e. cos^2 A

Since we have to get 2tan^2 A in the RHS, we need two sin^2 A in the numerator so, we expand 1 using the Trigonometric identity:
1 = sin^2 A + cos^2 A

And then we do simple division and match the terms to RHS.

Identity: tan x = sinx / cosx

So, this is the way to solve the Trigonometric identity of Trigonometry in mathematics.

Prove that: (1 - sinΘ)/ cosΘ = 1/ (secΘ + tanΘ)

Answer:

For solving this question we took the RHS because it was a lot easier to prove this Trigonometric identity through rhs.

First we take the RHS.

Then we change the terms into sin and cos.
i.e. sec x = 1/ cos x
tan x = sin x/ cos x

Then we applied LCM and using the various properties of ratios we solved the question!

Identity used: cos^2 x = 1 - sin^2 x

Therefore, we proved that the LHS is equal to the RHS.

Prove that: ( Cosec Θ - sin Θ) ( sec Θ - cos Θ) (tan Θ + cot Θ) = 1

Here, the first thing we did was we changed the all given terms into sin and cos.

I.e.
Cosec x = 1/ sin x
Sec x = 1/ cos x
Tan x= sin x/ cos x
Cot x= cos x/ sin x

And then we applied the LCM and tried to get everything in multiple form by eliminating the + or - signs which we changed using Trigonometric identity.

1 - sin^2 x = cos^2 x
1 - cos^2 x = sin^2 X

Then we had every thing in multiple forms and the numerator and denominator are equal. It means we get 1 while dividing the equal terms so our answer is 1 which is equal to the RHS.

Hence, we proved the Trigonometric identity of Trigonometry in mathematics.

Prove that: tan^2 Θ - sin^2 Θ = sin^2 Θ tan^2 Θ

Answer:

The process of solving this question and the important Trigonometric Ratios and other important stuffs are:

First we changed the given LHS in terms of sin and cos.

I.e. tan x = sin x / cos x

And then we took the LCM of the two terms.

Then we took the common term in the numerator and then wrote the remaining terms.

Then we got a Trigonometric identity
i.e. sin^2 x = 1 - cos^2 x

So, we got everything in the form of multiple and division.
Then we separated the terms to get sin and tan.

So, we.proved the given trigonometric identity of Trigonometry in mathematics which was a lot easier.

Prove that: Cot^2 A - cos^2 A = cos^2 A. Cot^2A

Answer:

The question follows a trick and with this trick you can solve every other questions just like this one.

First change the given terms into sin and cos

I.e. cot x = cos x/ sinx

And take the LCM

Then you will get similar terms in the numerator so take the common and you will get a trigonometric identity. Here we got

1 - sin^2 x = cos^2 x

And then you have everything in multiple and division form then you can simply convert them into cos and cot and prove that the given LHS is equal to the RHS and solve the Trigonometric identity of Trigonometry in mathematics.

Prove that:

sin^2 α. sec^2 β + tan^2 β. cos^2 α = sin^2α + tan^2 β

Answer:

To prove this Trigonometric identity, we need to understand that alpha and beta are not equal to each other unless they are said they are equal!

Now, first step we do is we change the given terms into sin and cos

Where,

Sec x = 1/ cos x
Tan x= sin x/ cos x

After this, take LCM, perform the operations and then take common ,cancel the equal terms of the denominator and numerator, if any. And then prove the LHS and RHS as shown in the image.

Prove that: tan^2 A + cot^2 A = sec^2 A.cosec^2 A - 2

Answer:

Here, to solve the question, we used the Trigonometric identity of tan A and cotA
Which is

Tan A = Sin A / Cos A

And

Cot A = Cos A / Sin A
After that we took the LCM and then opened up the algebraic formula.

Algebraic formulas used:
a^2 + b^2 = (a + b)^2 - 2 ab OR (a - b)^2 + 2 ab

We need to analyze which formula should be used here.
Let's see the RHS, we have (- 2)

This means we need to use the formula of:

a^2 + b^2 = (a + b)^2 - 2 ab

Then we expand the formula into it's factors and then use the Trigonometric identity:

Sin^2 x + cos^2 x = 1

And solved the question finally by changin
1/cos A = secA
And
1/ sinA = cosecA

So, you can prove this Trigonometric identity of Trigonometry using this process.
Prove that: sec^4 Θ + tan^4 Θ = 1 + 2tan^2 Θ sec^2 Θ

Answer:

This is a very simple Trigonometric question that you will ever get in Trigonometry. Here we have to use the algebraic formulas and Trigonometric identities.

Algebraic formulas used:
a^2 + b^2 = (a + b)^2 - 2 ab OR (a - b)^2 + 2 ab

We need to analyze which formula should be used here.
Let's see the RHS, we have (+ 2tan^2 Θ sec^2 Θ)

This means we need to use the formula of:

a^2 + b^2 = (a - b)^2 + 2 ab

And then we used the Trigonometric identity
I.e. sec^2 x - tan^2 x = 1

This way we can easily prove the given trigonometric identity.
Prove that: ( CosecΘ + 1) (1 - sinΘ) = cosΘ . cotΘ

Answer:

Here, the first thing I did is expand the expression. Then using the identity:

Cosec x * sin x = 1

And the I changed, Cosec x = 1/ sin x

Also, cos^2 x = 1 - sin^2 x

Then also we used, cot x = cos x / sin x

And then we can solve the question by showing the LHS is equal to the RHS.
Prove that: (1 + 3sin A - 4 sin^3 A) / ( 1- sinA) =< (1 + 2sinA)^2

Answer:

These questions are a medium difficulty questions and you need to apply the mid term factorization method to solve these questions.

First we expand the middle term into two numbers so that when we multiply those two numbers together we get the product of first and the last term. And when we put the operator of the last term in between those numbers, we get the middle term.

Here, first term = 1
Middle term = 3
Last term = 4

We have,
Number1 * Number2 = first term* last term
Or, 4*1 = 1*4

And, we have operator '-'
So, number1 - number2 = middle term
4-1= 3

So, we meet both the conditions when our first number is 4 and the second number is 1
Then we can expand the middle term.

Now, we can use the formula and factorization and canceling methods to prove as shown in the image above.
Prove that: (sin^3 A + cos^3 A) / ( cosA + sinA) = 1 - sinA cosA

Answer:

For solving this question we should remember two things!
First is: a^3 + b^3 = (a+b) (a^2 - ab + b^2)

And. Sin^2 A + cos^2 A = 1

After this it is too simple to show LHS is equal to the RHS.

Open the formula!
Use the identity!
Take common!
Cancel the equal expression from numerator and denominator.
Then you will get the LHS is equal to the RHS.
Prove that: sec^4 α - 1 = 2 tan^2 α + tan^4 α

Answer:

To prove: sec^4 α - 1 = 2 tan^2 α + tan^4 α

We need to analyze the LHS of the given question.

We have a^2 - b^2 which can be expanded as (a+b) ( a-b) which we did in the process.

Using identity: sec^2 x - tan^2 x = 1

And sec^2 x = 1 + tan^2 x

We solved the above question and proved that the LHS is equal to the RHS.
Prove that: ( 1 + cot Θ - Cosec Θ) (1 + tan Θ + sec Θ) = 2

Answer:

The simple way to solve this question is to change the given things into sin and cos respectively!

As we know,

CotΘ = cosΘ / sinΘ

CosecΘ = 1/ sinΘ

TanΘ = sinΘ / cosΘ

SecΘ = 1/ cosΘ

Then you can take the LCM and then start solving the question as shown in the photo.

Identity used: sin^2 Θ + cos^2 Θ = 1

See the image to visualize the answer!
Prove that:

(secA + tanA) / (secA - tanA) = 1 + 2 secA tanA + 2 tan^2 A

Answer:

Solving these questions are tricky! First you need to eliminate the denominator by changing the denominator into 1.

We know,
sec^2 A - tan^2 A = 1

Now, what do we have to do convert (sec A - tan A ) into sec^2 A - tan^2 A .
Wait I think we need to multiply by
(secA + tan A)

Because we know, (a - b) (a +b) = a^2 - b^2

But to put
(secA + tan A) we need to multiply and divide both by (secA + tan A).

I mean, (sec A + tan A) / (secA + tanA)

And we get the RHS.
Prove that:

(tan A + sin A) / (tan A - sin A) = (secA + 1) / (secA - 1)

Answer:

Being a beginner in Trigonometry, it is easier to solve the question such as this by converting the given Trigonometric Ratios into sin and cos as well as know,

TanA = sinA / cosA
SecA = 1 / cosA

Later on, we will take the LCM and then take the common in numerator and denominator and cancel the equal terms and then we will get the RHS.
Prove that: (sin A - cosA)^2 = 1 - 2.sinA.cosA

Answer:

The given question is too simple. We have been learning the formula of
(a-b)^2 = a^2 - 2 a b + b^2

Applying the same formula for the given question, we get:
(sinA - cosA)^2 = sin^2A -2 sinA cosA + cos^2A

And we have the identity of:

sin^2A + cos^2A = 1

So, we get 1 in place of sin^2A + cos^2A in
sin^2A -2 sinA cosA + cos^2A

So, 1 - 2 sinA cosA is what we have finally!

Therefore the LHS is equal to he RHS.
Prove that: 1 / ( secA - tanA) = secA + tanA

Answer:

When you see the questions having either
secA - tanA or secA + tanA on LHS or either of them in the RHS then remember you need to use the Trigonometric identity:

Sec^2 A - tan^2 A = 1

We have there, a^2 - b^2
Which we can open as the factor:
(a+ b) (a-b)
So,
(Sec A+ tanA) (secA- tanA)

Then we cancel the equal terms from the numerator and denominator.
And we get the LHS.
Prove that: cot^2 Θ - cos^2Θ = cot^2 Θ cos^2 Θ

Answer:

In the given question, it might feel that the identity is impossible to solve initially. But relax, think widely!

We know, cot^2 Θ = cos^2 Θ / sin^2 Θ

And when we take LCM after expanding
cot^2 Θ
We get something common in the numerator!
When we take the common term and write the remaining terms we get (1- sin^2 Θ) which is equal to cos^2 Θ [ an important Trigonometric identity] . Then it's easy to solve the question. See the picture above to visualize the answer now!
Prove that: (1 - sin Θ) / cos Θ = cos Θ / (1 + sin Θ)

Answer:

As you can already guess from the figure, the first step to solve such problems where you have completely different LHS and RHS. You need to rationalize the denominator and prove. Then we have the following identity:
Cos^2 x = 1 - sin^2 x

And yeah, this much explanation is ok. Now, check the figure with the answer to visualize the process!
Prove that: √ {(1- cos Θ) / (1 + cos Θ)} = cosec Θ - cot Θ

Answer:

The first step to solve such questions is not to panic!
When you see the question having square root then while proving the Trigonometric identities then remember you need to rationalize the denominator! Yes, the denominator has to be rationalized in most of the cases while we might need to rationalize the numerator in some cases as well!

Next step is to open the formulas and turn them into that term which can be easily cancel the square root and can be rational number!

After that, we can easily solve the question by applying Trigonometric identities and tricks!

Here we used,

Cosec Θ = 1 / sin Θ and Cot Θ = cos Θ / sin Θ
Prove that: √{(1 - sinB) / (1 + sinB)} = secB + tanB

The first step to solve such questions is not to panic!
When you see the question having square root then while proving the Trigonometric identities then remember you need to rationalize the denominator! Yes, the denominator has to be rationalized in most of the cases while we might need to rationalize the numerator in some cases as well!

Next step is to open the formulas and turn them into that term which can be easily cancel the square root and can be rational number!

After that, we can easily solve the question by applying Trigonometric identities and tricks!

Here we used,

SecB = 1/ cosB and tanB = sinB / cosB
Prove that: √{(1 - sinA ) / (1 + sinA)} = secA + tanA

Answer:

The explanation has already been given in the figure step by step!

We used the identity:
secA = 1 / cosA
And
tanA = sinA/ cosA

Also here, we rationalized the given denominator to get the term we needed!
Prove that: 1 - ( cos^2 Θ / 1 + sin Θ) = sin Θ

Answer:

Brief explanation:

We took the LCM of the two terms in the first step.
Then, we open up the formula. I.e. (a+b)(a-b) = a^2-b^2
In the next step, we get terms common which we put in the front of two terms and put remaining values of those two terms.
Then we cancel the equal terms from the numerator and denominator.
Then we expand the expression and we get the R.H.S.
Prove that:

sin α / (1 + cos α) + (1 + cos α) / sin α = 2 cosec α

Answer:

Explanation;

Step1: We wrote the same question taking the L.H.S. ( Left Hand Side).

Step2: We took the L.C.M. (Lowest Common Multiple).

Step3: We put them into the formula.

Step4: We opened the formula. i.e. (a+b)^2 = a^2 + 2ab + b^2

Step5: then we use the identity: sin^2a + cos^2a = 1

Then taking the terms common in numerator and denominator, we cancel the equal terms and then we have Cosec a = 1 / sin a

So, this is how we prove the given identity.
Prove the following identity:
(sin^3θ - cos^3θ) / (sinθ - cosθ) = (1+sinθcosθ)

We have to use the factorization formula of a^3 - b^3 which is (a-b) (a^2 +ab +b^2)

The solved answer of this question is attached here.

I hope that I have been able to clarify your doubt on this question. It's simple. Just analyze the questions and figure out the possible way to solve it.
Prove the following identity:

cosθ / (1-sinθ) - cosθ / (1+sinθ) = 2 tanθ

I have solved the question and I am going to attach the answer here.

This is the process how you solve the given trigonometric identity. Make sure to take the LCM and then start solving. Remember the identities and formulas that we learn previously in previous blogs.