Answer
$\cos (5t)+\cos (3t)=2\cos 4t\cos t.$
Work Step by Step
Consider the function,
\begin{align*}
\cos (5t)+\cos (3t).
\end{align*}
According to the formula for sum o two cosines, we have
\begin{align*}
\cos A+ \cos B=2\cos \dfrac{A+B}{2}\cos \dfrac{A-B}{2}.
\end{align*}
Let $A=5t, B=3t$. Thus, we get
\begin{align*}
\cos 5t+ \cos 3t&=2\cos \dfrac{5t+3t}{2}\cos \dfrac{5t-3t}{2}\\
&=2\cos \dfrac{8t}{2}\cos \dfrac{2t}{2}\\
&=2\cos 4t\cos t.
\end{align*}
Hence, $\cos (5t)+\cos (3t)=2\cos 4t\cos t.$
