how to prove (1-tan^2*2x/1+tan^2*2x) = 2cos4x

Question

I really don´t know how to even start proving this problem.

Answer 1

Let x=π/6, 2x=π/3 and tan(π/3)=√3. cos(4x)=cos(2π/3)=-½. 1+tan²(2x)=1+3=4; 1-tan²(2x)=1-3=-2.

Therefore (1-tan²(2x))/(1+tan²(2x))=-2/4=-½=cos(4x), not 2cos(4x).

1+tan²A≡sec²A=1/cos²A (trig identity) for all angles A. Therefore the denominator becomes sec²(2x)=1/cos²(2x).

We multiply the numerator by cos²(2x) to get cos²(2x)-sin²(2x)≡cos(2×2x)=cos(4x) (trig identity: cos(2A)=cos²A-sin²(A)).

[2cos(4x)=2cos²(2x)-2sin²(2x).]