Trigonometric Functions of Sum and Difference of Two Angles


  • Identities Related to Sin 2x, Cos2x, Tan 2x, Sin3x, Cos3x and Tan3x.
  • Deducing the Identities
  • Deducing the identities like the following:-
    `tan(x+-y)=(tanx+-tany)/(1+-tanxtany)", "cot(x+-y)=(cotxcoty+-1)/(coty+-cotx)`


1) sin (-x)= -sin x

2)  cos (–x) = cos x

3) sin (x+y)= sin x cos y+ cos x sin y

4) sin (x-y)= sin x cos y- cos x sin y 

5) cos (x+y)= cos x cos y- sin x sin y

6) cos (x-y)= cos x cos y+ sin x sin y

7) cos `(π/2 - x)`= sin x

8) cos `(π/2 +x)`= -sin x

9) cos (π- x)= -cos x

10) cos (π+ x)= -cos x

11) cos (2π- x)= cos x

12) sin `(π/2 - x)`= cos x

13) sin `(π/2 + x)`= cos x

14) sin (π- x)= sin x

15) sin (π+ x)= -sin x

16) sin (2π- x)= -sin x

17) tan (x+y)= `(tan x+ tan y)/(1- tan x tan y)`

18) tan (x-y)= `(tan x- tan y)/(1+ tan x tan y)`

19) cot (x+y)= `(cot x cot y- 1)/(cot y+ cot x)`

20) cot (x-y)= `(cot x cot y+1)/cot y- cot x`

21) sin2x= 2 sin x cos x

      sin2x= `(2 tan x)/(1+ tan^2x)`

22) cos2x= `cos^2 x- sin^2 x`

     cos2x= `2 cos^2 x- 1`

       cos2x= `1- 2 sin^2 x`

       cos2x= `(1- tan^2 x)/(1+ tan^2 x)`

23) tan2x= `(2 tan x)/(1- tan^2 x)`

24) sin3x= `3 sin x- 4 sin^3 x`

25) cos3x= `4 cos^3 x- 3 cos x`

26) tan3x= `(3 tan x- tan^3 x)/(1- 3 tan^2 x)`

27) cos x+ cos y= `2cos  (x+y)/2 cos  (x-y)/2`

28) cos x- cos y= `-2 sin  (x+y)/2 sin  (x-y)/2`

29) sin x+ siny= `2 sin  (x+y)/2 cos  (x-y)/2`

30) sin x- sin y= `2 cos  (x+y)/2 sin  (x-y)/2`

31) sin (x+y)+ sin(x-y)= 2 sin x cos y

32) sin (x+y)- sin (x-y)= 2 cos x sin y

33) cos (x+y)+ cos (x-y)= 2 cos x cos y

34) cos (x+y)- cos (x-y)= 2 sin x sin y

35) 2 cos x cos y = cos (x + y) + cos (x – y)

36) –2 sin x sin y = cos (x + y) – cos (x – y)

37) 2 sin x cos y = sin (x + y) + sin (x – y)

38) 2 cos x sin y = sin (x + y) – sin (x – y)

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