Answer
The value of $10{{\cos }^{4}}x$ is $\frac{15}{4}+5\cos 2x+\frac{5}{4}\cos 4x$.
Work Step by Step
We have to find the value of $10{{\cos }^{4}}x$; various cos formulas are used which represent the answer in the form of cos.
$\begin{align}
& 10{{\cos }^{4}}x=10{{\cos }^{2}}x{{\cos }^{2}}x \\
& =10{{\left( \frac{1+\cos 2x}{2} \right)}^{2}} \\
& =10\left( \frac{1+2\cos 2x+{{\cos }^{2}}2x}{4} \right) \\
& =\frac{10+20\cos 2x+10{{\cos }^{2}}2x}{4}
\end{align}$
And take the denominator of the equation with each different numerator.
$\begin{align}
& 10{{\cos }^{4}}x=\frac{10}{4}+\frac{20\cos 2x}{4}+\frac{10{{\cos }^{2}}2x}{4} \\
& =\frac{5}{2}+5\cos 2x+\frac{5}{4}\left( 1+\cos 4x \right) \\
& =\frac{5}{2}+5\cos 2x+\frac{5}{4}+\frac{5}{4}\cos 4x \\
& =\frac{15}{4}+5\cos 2x+\frac{5}{4}\cos 4x
\end{align}$
So, in step 2 the identity of ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ is used and in the last step, $\frac{5}{4}$ is distributed and added into $\frac{5}{2}$.
Thus, the value of $10{{\cos }^{4}}x$ is $\frac{15}{4}+5\cos 2x+\frac{5}{4}\cos 4x$.
