Trigonometrical Identities

1. Quotient relations:

  1. tan = sin /cos
  2. cot = cos /sin

2. Reciprocal relations:

  1. sin = 1/ cosec
  2. cosec = 1/sin
  3. cos = 1/sec
  4. sec = 1/cos
  5. tan = 1/cot
  6. cot = 1/tan

3. Square relations (Fundamental Identities):

  1. sin² +cos² = 1
  2. 1 +tan² = sec²
  3. 1 +cot² = cosec²

4. T-ratios of standard angles:

  0° 30° 45° 60° 90°
Sin
Cos

5. T-ratios of complementary angles:

  1. sin (90° - ) = cos
  2. cos (90° - ) = sin
  3. tan (90° - ) = cot
  4. cot (90° - ) = tan
  5. sec (90° - ) = cosec
  6. cosec (90° - ) = sec

Exercise

Prove the following identities (1 -15):

  1. tan² - 1/cos² +1 = 0
  2. sin A/(1 +cosA) +(1 +cos A)/sin A = 2 cosecA
  3. (1 -sin )/(1 +sin ) = (sec -tan )²
  4. sec² A +cosec² A = sec² A cosec²A
  5. sec4 -tan4 = 1 +2 tan²
  6. tan² A/(1+tan² A) + cot² A/(1 +cot² A) = 1
  7. (1 -cos )(1 +sec ) = tan sin
  8. (cot² A -tan² A)/(cot A +tan A)² = 2 cos² A -1
  9. cos4 +sin4 +2 sin² cos² = 1
  10. = cosec +cot
  11. .
  12. sec (1 -sin )(sec +tan ) =1
  13. (sec A +cos A)(sec A -cos A) = tan²A +sin² A
  14. tan² -sin² = tan² sin²
  15. (1 -cos )(1 + cos )(1 +cot² ) = 1

Simplify the expression in questions (16 -27):

  1. (sin² A -cos² A)/(sin A -cos A)
  2. sin² cos cosec³ sec
  3. cot B sin² B cot B
  4. (cos² A +cos A -12)/cos A - 3)
  5. tan /(sec -1) + tan /(sec +1)
  6. sec A csc A -tan A -cot A
  7. (cot² +tan² )/(cos² sec² )
  8. x = a sec , y = b tan
  9. x = h +a cos , y = k +b sin
  10. x = a sec³ , y = b tan³
  11. tan +sin = m, tan - sin = n
  12. cot +cos = m, cot -cos = n

When 0° < < 90°, solve the following equations (28-33):

  1. 2 sin² = 1/2
  2. 4 cos² -3 = 0
  3. sin² -(1/2)sin = 0
  4. tan² = 3 (sec -1)
  5. 2 cosec = 3 sec²
  6. 3 tan +cot = 5 cosec

Without using trigonometric tables, evaluate (34-43):

  1. (cos0° +sin45° +sin30°).(sin90° -cos45° +cos60°)
  2. sin² 45° -tan² 60° + cos² 90°
  3. sin 23°/cos 67°
  4. cosec 31°/ sec 59°
  5. sin² 38° -sin²52°
  6. sin 18° -cos 72°
  7. sin 36° sec 54° +cos 24° cosec 66°
  8. .
  9. cosec²67° -tan² 23°
  10. sec 31° sin 59° +cos 31° cosec 59°

Express the following in terms of t-ratios of angles between 0° and 45°.

  1. sin 85° +cosec 85°
  2.  cosec 69° +cot 69°
  3. sin 81° +tan 81°
  4. cos 56° +cot 56°

Prove the following:

  1. [sin (90 -A) sin A]/tan A-1 = - sin² A
  2. cos cos(90° - ) -sin sin (90° - ) = 0
  3. sin (90° - ) cos (90° - ) = tan /(1 +tan² )
  4. cosec² (90° - ) -tan²   = cos²(90° - ) +cot²
  5. If cos/cos = m and cos/sin = n, show that (m² +n²) cos² = n².
  6. If x = r cos sin, y = r cos cos and z = r sin , show that x² +y² +z² = r².

Answers

16. sin A +cos A        17. csc       18. cos² B          19. cos A +4
20. 2 csc                   21. 0            22. 1                  23. x²/a² - y²/b² = 1
24.           25.
26. (m² -n²)² = 16 mn                    27. (m² -n²)² = 16 mn
28. 30°                       29. 30°        30. 30°
31. 60°                       32. 30°        33. 60°
34. -5/2                      35. 1            36. 1
37. 1                           38. 0           39. 2                 40. 1/2
41. 1                           42. 2             43. 0
44. sec 21° +tan 21°                       45. cos 9° +cot 9°
46. sin 34° +tan 34°                        47. sin²