rove 2 SolutionTo prove Sin 4 2x 3812Cos 4x18Cos 8x RHS 18

Solution

To prove :

Sin ⁴ (2x)=( 3/8)-(1/2)Cos (4x)+(1/8)Cos (8x)

R.H.S. =(1/8)( 3- 4Cos (4x)+Cos 8x)

Cos 4x= 1-2 Sin ² (2x) ; here we used two trigonometric identities cos² x = 1-Sin²x and Cos(2x) = Cos²x - Sin²x

Similary, using the above identity; Cos 8x= 2 cos² (4x) -1= 2 ( 1-2 Sin² ( 2x))² -1

= 2 (1+4 Sin ⁴ (2x) -4 Sin² (2x))-1

R.H.S = (1/8)(3- 4+8Sin² (2x) + 2+8Sin⁴ (2x)

-8 Sin² (2x)-1)

R.H.S. = (1/8)(8 Sin ⁴ (2x)) = Sin⁴ (2x) hence proved.