# How do you verify 8sin²xcos²x = 1-cos^4(x)?

##### 1 Answer
Jun 3, 2015

This can not be proven because it is not in general true.

As a counter example, consider $x = \frac{\pi}{4}$

$\textcolor{w h i t e}{\text{XXXX}}$$\sin \left(x\right) = \frac{1}{\sqrt{2}}$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$${\sin}^{2} \left(x\right) = \frac{1}{2}$
$\textcolor{w h i t e}{\text{XXXX}}$$\cos \left(x\right) = \frac{1}{\sqrt{2}}$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$${\cos}^{2} \left(x\right) = \frac{1}{2}$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$${\cos}^{4} \left(x\right) = \frac{1}{4}$

$8 {\sin}^{2} x {\cos}^{2} x = 8 \cdot \frac{1}{2} \cdot \frac{1}{2} = 2$

$1 - {\cos}^{4} \left(x\right) = 1 - \frac{1}{4} = \frac{3}{4}$

$8 {\sin}^{2} x {\cos}^{2} x \ne 1 - {\cos}^{4} x$