Question

Use Pythagorean identities to rewrite each of the following trigonometric expressions.

${\cos }^{4}x=\_\_\_\_\_$

Short answer

The obtained result is$\begin{array}{rcl}& & 1+{\sin }^{4}x-2{\sin }^{2}x\end{array}$.

Step-by-step solution

Step 1. Given information.

Consider the given expression.

${\cos }^{4}x$

Step 2. Apply the Pythagorean identities to rewrite the expression.

The Pythagorean identity is defined as,

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

Use the identity to write the given expression.

localid="1651821571277" role="math" $\begin{array}{rcl}{\sin }^{2}x+{\cos }^{2}x& =& 1\\ {\cos }^{2}x& =& 1-{\sin }^{2}x\\ {\left({\cos }^{2}x\right)}^2& =& {\left(1-{\sin }^{2}x\right)}^2\\ {\cos }^{4}x& =& 1+{\sin }^{4}x-2{\sin }^{2}x\end{array}$