Answer
$cos3\theta=4cos^{3}\theta-3cos\theta$
Work Step by Step
We need to prove the identity
$cos3\theta=4cos^{3}\theta-3cos\theta$
Now, $cos3\theta$ can be written as:
$cos3\theta=cos(2\theta+\theta)$
Use sum identity for cosine.
$cos 3\theta=cos(2\theta+\theta)=cos2\theta cos\theta-sin2\theta sin\theta$
$=(2cos^{2}\theta -1)cos\theta-(2sin\theta cos\theta ) sin\theta$
$=2cos^{3}\theta - cos\theta-2sin^{2}\theta cos\theta$
$=2cos^{3}\theta - cos\theta-2(1-cos^{2}\theta) cos\theta$
$=2cos^{3}\theta - cos\theta-2 cos\theta+2cos^{3}\theta $
Hence, $cos3\theta=4cos^{3}\theta-3cos\theta$
