Question

Prove:
1. (cos^2⁡ x-1)^2=(1-2 cos⁡2x+cos^2⁡ 2x)/4
2.tan⁡(π+4x)=(4 tan⁡x-4 tan^3⁡ x)/(tan^4 ⁡x-6 tan^2 ⁡x+1)

Answer 1

1. LS = (cos^2 x - 1)^2
= (-sin^2 x)^2
= sin^4 x

RS = (1-2 cos⁡2x+cos^2⁡ 2x)/4
= (1 - cos^2 (2x))^2 / 4
= (1 - (1 - 2sin^2)^2 / 4
= (2sin^2 x)^2 / 4
= 4sin^4 x / 4
= sin^4 x
= LS

What is your progress for #2?
You might want to start with tan(π + 4x) = tan (4x)
= tan(2x + 2x) = 2 tan(2x)/(1 - tan^2 (2x) )
and again: tan (2x) = 2tanx/(1 - tan^2 x)

see what you can with that.

Answer 2

Prove: tan⁡(π+4x)=(4 tan⁡x-4 tan^3⁡ x)/(tan^4 ⁡x-6 tan^2 ⁡x+1)