Question

Stuck, help please !

Answer 1

Use the fundamental trigonometric equation

$\sin^2(x) + \cos^2(x) = 1 \implies \sin^2(x) = 1-\cos^2(x)$

To rewrite the equation as

$1-\cos^2(x) = 1-\cos(x)$

Subtract 1 from both sides:

$-\cos^2(x) = -\cos(x)$

Add $\cos^2(x)$ to both sides:

$\cos^2(x) - \cos(x) = 0 \iff \cos(x)\left(\cos(x)-1\right)=0$

A multipication is zero if and only if one of the factors is zero. So, the solutions are given by

$\cos(x)=0 \iff x=\dfrac{\pi}{2}+k\pi,\quad k \in \mathbb{Z}$

or

$\cos(x)-1=0 \iff \cos(x)=1 \iff x=2k\pi,\quad k \in \mathbb{Z}$