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Mathematics

Solved Exercises

- Class 09



Unit-4 - Trigonometry

Trigonometric Identities

  1. Prove that: cosθ / (1-sinθ) - cosθ / (1+sinθ) = 2 tanθ
  2. Prove that: (sin^3θ - cos^3θ) / (sinθ - cosθ) = (1+sinθcosθ)
  3. Prove that: sin α / (1 + cos α) + (1 + cos α) / sin α = 2 cosec α
  4. Prove that: 1 - ( cos^2 Θ / 1 + sin Θ) = sin Θ
  5. Prove that: 1 / ( secA - tanA) = secA + tanA
  6. Prove that: cot^2 Θ - cos^2Θ = cot^2 Θ cos^2 Θ
  7. Prove that: (1 - sin Θ) / cos Θ = cos Θ / (1 + sin Θ)
  8. Prove that: √ {(1- cos Θ) / (1 + cos Θ)} = cosec Θ - cot Θ
  9. Prove that: √{(1 - sinB) / (1 + sinB)} = secB + tanB
  10. Prove that: √{(1 - sinA ) / (1 + sinA)} = secA + tanA
  11. Prove that: (sin^3 A + cos^3 A) / ( cosA + sinA) = 1 - sinA cosA
  12. Prove that: sec^4 α - 1 = 2 tan^2 α + tan^4 α
  13. Prove that: ( 1 + cot Θ - Cosec Θ) (1 + tan Θ + sec Θ) = 2
  14. Prove that: (secA + tanA) / (secA - tanA) = 1 + 2 secA tanA + 2 tan^2 A
  15. Prove that: (tan A + sin A) / (tan A - sin A) = (secA + 1) / (secA - 1)
  16. Prove that: (sin A - cosA)^2 = 1 - 2.sinA.cosA
  17. Prove that: Cot^2 A - cos^2 A = cos^2 A. Cot^2A
  18. Prove that: sin^2 α. sec^2 β + tan^2 β. cos^2 α = sin^2α + tan^2 β
  19. Prove that: tan^2 A + cot^2 A = sec^2 A.cosec^2 A - 2
  20. Prove that: sec^4 Θ + tan^4 Θ = 1 + 2tan^2 Θ sec^2 Θ
  21. Prove that: ( CosecΘ + 1) (1 - sinΘ) = cosΘ . cotΘ
  22. Prove that: (1 + 3sin A - 4 sin^3 A) / ( 1- sinA) =< (1 + 2sinA)^2
  23. Prove that: (tanΘ + cotΘ)² = (1 + tan²Î˜) + (1 + cot²Î˜)
  24. Prove that: (sinΘ - cosΘ + 1) (sinΘ + cosΘ - 1) = secΘ + tanΘ
  25. Prove that: (1 - sin^4 A) / cos^4 A = 1 + 2 tan^2 A
  26. Prove that: (1 - sinΘ)/ cosΘ = 1/ (secΘ + tanΘ)
  27. Prove that: ( Cosec Θ - sin Θ) ( sec Θ - cos Θ) (tan Θ + cot Θ) = 1
  28. Prove that: tan^2 Θ - sin^2 Θ = sin^2 Θ tan^2 Θ
  29. Prove that: (tanΘ +secΘ -1) / (tanΘ -secΘ +1) = (1+ sinΘ)/cosΘ = secΘ + tanΘ
  30. Prove that: tan²A/ (tanA -1) + cotA /(1 - tanA) = 1 + secA.cosecA
  31. Prove that: (3 - 4 sin^2 α) /(1- 3tan^2 α) = (3 -tan^2 α) (4cos^2 α -3)
  32. Prove that: (cosA - sinA +1)/ (cosA +sinA -1) = cotA + cosecA
  33. Prove that: tan x/ (secx - 1) - sinx/ (1 + cosx) = 2 cotx
  34. Prove that: (cosA - sinA + 1) / (cosA + sinA +1) = (1 - sinA) / cosA
  35. Prove that: tanα / (1 +tan^2 α)^2 + cotα / (1+ cot^2 α)^2 = sinα. Cosα
  36. Prove that: (cotA + cosecA -1)/ (cotA -cosecA +1) = (1 +cosA)/ sinA
  37. Prove that: (1 + tanΘ)^2 + (1 + cotΘ)^2 = (secΘ + cosecΘ)^2
  38. Prove that: 1 / (secA -tanA) - 1/ cosA = 1/ cosA - 1/(secA + tanA)
  39. Prove that: If cos^4 A + cos^2 A =1, then prove that: tan^4 A+ tan^2 A =1