1. ## [SOLVED] solve equation

sin2theta-1=cos2theta

2. Originally Posted by Jaffa
sin2theta-1=cos2theta
$\displaystyle \sin 2 \theta-1=\cos 2 \theta$

Substitute double angle identities:

$\displaystyle \boxed{\sin 2 \theta = 2 \sin \theta \cos \theta}$

$\displaystyle \boxed{\cos 2 \theta = 1 - 2 \sin^2 \theta}$

$\displaystyle 2 \sin \theta \cos \theta - 1 = 1 - 2 \sin^2 \theta$

$\displaystyle 2 \sin^2 \theta + 2 \sin \theta \cos \theta-2=0$

$\displaystyle {\color{red}\sin^2 \theta}+ \sin \theta \cos \theta {\color{red}- 1} = 0$

Identity: Since $\displaystyle \sin^2 \theta+ \cos^2 \theta=1$, then
$\displaystyle \boxed{{\color{red}\sin^2 \theta - 1 = -\cos^2 \theta}}$

$\displaystyle -\cos^2 \theta+ \sin \theta \cos \theta=0$

$\displaystyle \cos^2 \theta-\sin \theta \cos \theta=0$

$\displaystyle \cos \theta(1-\sin \theta)=0$

$\displaystyle \boxed{\cos \theta = 0} \ \ or \ \ 1-\sin \theta=0$

$\displaystyle -\sin \theta = -1$

$\displaystyle \boxed{\sin \theta=1}$

$\displaystyle \cos \theta=0 \ \ at \ \ \left\{\frac{\pi}{2}, \frac{3\pi}{2}\right\}$

$\displaystyle \sin \theta = 1 \ \ at \ \ \left\{\frac{\pi}{2}\right\}$

Solution set = $\displaystyle \left\{\frac{\pi}{2}, \frac{3\pi}{2}\right\}$