Trig Equations with Multiple Trig Functions help

$\tan x = \cos x$
$\tan x - \cos x = 0$
$\frac{\sin x}{\cos x} - \cos x = 0$
$\sin x - \cos^{2} x = 0 \quad \quad \mbox{Multiply both sides by } \cos x$
$\sin x - \left(1 - \sin^{2} x\right) = 0$
$\sin^{2} x + \sin x - 1 = 0$

Solve the quadratic for sin x and solve for x.

-----------------

$\cos \left(\frac{1}{2}x\right) = \cos x$
$\cos^{2}\left(\frac{1}{2}x\right) = \cos^{2} x$
$\frac{1}{2} \left(1 + \cos x\right) = \cos^{2} x \quad \quad \mbox{Half-angle Identity}$
$0 = \cos^{2} x - \frac{1}{2} \cos x - \frac{1}{2}$
$0 = 2 \cos^{2} x - \cos x - 1$

You know the dril.