Answer
To reduce the power of $\left( {{\cos }^{4}}x \right)$, first rewrite $\left( {{\cos }^{4}}x \right)$ as ${{\left( {{\cos }^{2}}x \right)}^{2}}$. Then expand $\left( {{\cos }^{2}}x \right)$ using power reducing formula. Replace the term $\left( {{\cos }^{2}}x \right)$ in expression ${{\left( {{\cos }^{2}}x \right)}^{2}}$ by the result obtained from power reducing formula and expand it. While expanding there will be another term $\left( {{\cos }^{2}}2x \right)$. Express this term also using power reducing formula. Final result is the power reduced form of ${{\cos }^{4}}x$.
Work Step by Step
Now, rewrite $\left( {{\cos }^{4}}x \right)$ as ${{\left( {{\cos }^{2}}x \right)}^{2}}$. Then expand $\left( {{\cos }^{2}}x \right)$ by using the power reducing formula:
$\begin{align}
& {{\cos }^{4}}x={{\left( {{\cos }^{2}}x \right)}^{2}} \\
& {{\cos }^{2}}x=\frac{1+\cos 2x}{2} \\
\end{align}$
So,
$\begin{align}
& {{\cos }^{4}}x={{\left( {{\cos }^{2}}x \right)}^{2}} \\
& ={{\left( \frac{1+\cos 2x}{2} \right)}^{2}} \\
& =\frac{1}{4}{{\left( 1+\cos 2x \right)}^{2}}
\end{align}$
Expand ${{\left( 1+\cos 2x \right)}^{2}}$ using ${{\left( a+b \right)}^{2}}.$ This means:
${{\left( 1+\cos 2x \right)}^{2}}=1+2\cos 2x+{{\cos }^{2}}2x$
Now, replace $x\text{ by }2x$ in $\left( {{\cos }^{2}}x=\frac{1+\cos 2x}{2} \right)$ to get $\left( {{\cos }^{2}}2x \right)$ That means:
$\begin{align}
& {{\cos }^{2}}2x=\frac{1+\cos 2.2x}{2} \\
& =\frac{1+\cos 4x}{2}
\end{align}$
Use ${{\cos }^{2}}2x=\frac{1+\cos 4x}{2}$ to expand $\left( 1+2\cos 2x+{{\cos }^{2}}2x \right).$ That is:
$\begin{align}
& 1+2\cos 2x+{{\cos }^{2}}2x=1+2\cos 2x+\frac{1+\cos 4x}{2} \\
& =\frac{2+4\cos 2x+1+\cos 4x}{2} \\
& =\frac{3+4\cos 2x+\cos 4x}{2} \\
& {{\left( 1+\cos 2x \right)}^{2}}=\frac{3+4\cos 2x+\cos 4x}{2}
\end{align}$
Thus,
$\begin{align}
& {{\cos }^{4}}x=\frac{1}{4}{{\left( 1+\cos 2x \right)}^{2}} \\
& =\frac{3+4\cos 2x+\cos 4x}{4.2} \\
& =\frac{3+4\cos 2x+\cos 4x}{8}
\end{align}$
