50 common Trigonometric Identities

For competitive exam of CSS/ FPSC/ NTS/ PMS/ PAF/ AFNS mathematics is included compulsory as well as optional subject for some tests. All competitive tests are scheduled for 2023. As CSS exam will held in October (MPT) & NTS/ AFNS in August. Candidates applying for any of these competitive exams 2023, must learn these basic trigonometric signs.

As mostly past papers are repeated which may prove as Game changers for the qualifying candidates to assure their success. All participating candidates must have to know these terms for Mathematics MPT or other tests.

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Trigonometric Signs/ Identities

Shifting Properties sum to product formuls
1. sin(-x) = -sin(x)29. sin(x)±sin(y)=sin(x±y/2)cos(x-y/2)
2. cos(-x) = cos(x)30. cos(x)+cos(y)= cos (x+y/2)cos(x-y/2)
3. sin(π/2-x) = cos(x)31. cos(x)-cos(y)= 2sin(x=y/2)sin(y-x/2)
4. cos(π/2-x) = sin(x) product to sum formulas
5. sin(π/2+x) = cos(x)32. 2sin(x)cos(y)= sin(x+y)+sin(x-Y)
6. cos(π/2+x) = -sin(x)33. 2sin (x)sin(y)= cos(x-y)-cos(x+y)
7. sin(π-x) = sin(x)34. 2cos(x)cos(y)= cos(x-y)+cos(x+y)
8. cos(π-x) = -cos(x)Half angle formulas
9. sin(π+x) = -sin(x)35. tan(x/2)=sin(x)/1+cos(x)=1-cos(x)/sin(x)
10. cos(π+x) = -cos(x)36. cos (x/2)= 1+cos(x)/2
11. sin(2π-x) = -sin(x)37. sin(x/2)= 1-cos(x)/2
12. cos(2π+-x) = cos(x)Power reduction formulas
13. sin(3π/2+-x) = -sin(x) 38. sin² x= 1/2 – cos2x/2
14. cos(3π/2+-x) = +-sin(x)39. cos² x= 1/2 + cos2x/2
Addition & Subtraction formulas 40. sin³ x= 3sinx/4 + sin3x/4
15. sin(x+-y) = sin(x)cos(y)+-cos(x)sin(y)41. cos³ x= 3cosx/4 + cos3x/4
16. cos(x+-y) = cos(x)cos(y) +- sin(x)sin(y)42. sin⁴ x= 3/8 – cos2x/4 +cos4x/8
17. tan(x+-y) = tan(x) +- tan(y)/1- +/- tan(x)tan(y)43. cos⁴  x= 3/8 + cos2x/2 + cos4x/8
18. cot(x+-y) = cot(x)cot(y)+-1/cot(y)+-cot(x)Parametric representation of trigonometric functions used for solving trig equations as polynomials
Multiple Angle Formulas 44. sin(x)= 2t/1+t² cos(x)= 1-t² /1+t²
19. sin(2x) = 2sin(x)cos(x)45. tan(x)= 2t/1-t² tan(x/2)=t
20. cos(2x) = cos square (x) – sin2(x) = 2cos square(x)-1 = 1-2sinSquare (x) Misc Formulas
21. cos(2x) = 1- tan square (X)/1+ tan sq (x) sin(2.x)= 2tan(x)/ 1+ tan square x46. sin4x= 4sinx cosx – 8sin³x cos³x
22. tan2x = 2tan(x)/1-tan sq (x)47. cos4x= 8cos⁴ x-8cos²x+1
23. sin3x= 3sin(x)-4sin³ (x)48. sinx+cosx= 2 sin(x+π/4)=2 cos(x-π/4)
24. cos3x= 4cos³(x)-3cos(x)
25. tan3x= 3tan(x)-tan³(x)/1-3tan³x
26. sin3x= 4sin(x)sin(π/3-x)sin(π/3+x)
27. cos3x= 4cos(x)cos(π/3-x)cos(π/3+x)
28. tan3x= tan(x)tan(π/3-x)tan(π/3+x)

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