# Cos(x)=sin(x). Then the value of cos^4x + sin^4x=?

May 14, 2018

$\frac{1}{2}$

#### Explanation:

$C o s \left(x\right) = \sin \left(x\right)$

Usually I avoid dividing by $\sin x$ or $\cos x$ because I lose solutions, but we will try solving the equation both ways to see if we get the same solution and all solutions are included:
Divide by $\sin x$ on both sides:
$\cot x = 1$
$x = \frac{\pi}{4} , \frac{5 \pi}{4}$
$x = \frac{\pi}{4} + \pi n$

Divide by $\cos x$ on both sides:
$\tan x = 1$
$x = \frac{\pi}{4} , \frac{5 \pi}{4}$
$x = \frac{\pi}{4} + \pi n$

Yields the same solution either way so:
${\left(\pm \frac{\sqrt{2}}{2}\right)}^{4} + {\left(\pm \frac{\sqrt{2}}{2}\right)}^{4} =$

$\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$