Question

Cos^2x-sin^2x=sqrt 3/2

Answer 1

$\cos^2x-\sin^2x=\cos2x=\dfrac{\sqrt3}2$

In the interval $[0,2\pi)$, $\cos x$ takes on the value of $\dfrac{\sqrt3}2$ when $x=\pm\dfrac\pi3$. In general, this happens for $x=\pm\dfrac\pi3+2n\pi$ where $n$ is any integer.

So, the solutions to this equation satisfy

$2x=\pm\dfrac\pi3+2n\pi$

or, dividing by 2,

$x=\pm\dfrac\pi6+n\pi$

for all integers $n$.