Answer
$$\cos 3 \theta =4 \cos ^{3} \theta-3 \cos \theta$$
Work Step by Step
We use the know identities to obtain:
\begin{aligned}
\cos 3 \theta &=\cos (2 \theta+\theta)\\
&=\cos 2 \theta \cos \theta-\sin 2 \theta \sin \theta\\
&=\left(2 \cos ^{2} \theta-1\right) \cos \theta-(2 \sin \theta \cos \theta) \sin \theta \\
&=\cos \theta\left(2 \cos ^{2} \theta-1-2 \sin ^{2} \theta\right)\\
&=\cos \theta\left(2 \cos ^{2} \theta-1-2\left(1-\cos ^{2} \theta\right)\right) \\
&=\cos \theta\left(2 \cos ^{2} \theta-1-2+2 \cos ^{2} \theta\right)\\
&=4 \cos ^{3} \theta-3 \cos \theta
\end{aligned}
