Question

Express ${\sin }^{4}x$ in terms of $cosine$ functions

Answer 1

We know that,

$\cos 2x=1-2{\sin }^{2}x$

$\Rightarrow 2{\sin }^{2}x=1-\cos 2x$

$\Rightarrow {\sin }^{2}x=\frac{1-\cos 2x}{2}$

Squaring both side, and we get

$\Rightarrow {\left({\sin }^{2}x\right)}^2={\left(\frac{1-\cos 2x}{2}\right)}^2$

$\Rightarrow {\sin }^{4}x=\frac{1+{\cos }^{2}2x-2\cos 2x}{4}$

Hence, this is the answer.