Show that \( \cos 3t = 4 \cos^3 t - 3 \cos t \)

5th Ed: #21

First we can apply the sum formula for cosine,

\( \cos 3t = \cos \left ( 2t+t \right ) = \cos \left ( 2t \right ) \cos \left ( t \right ) - \sin \left ( 2t \right ) \sin \left ( t \right ) \)

Now we will apply the sum formula to both sine and cosine,

\( = ( \cos t \cos t - \sin t \sin t )\cos t - (\sin t \cos t + \cos t \sin t ) \sin t \)
\( = \cos ^3 t - \cos t \sin ^2 t - 2 \sin ^2 t \cos t \)
\( = \cos ^3 t -\sin ^2 t ( \cos t + 2 \cos t ) \)
\( = \cos ^3 t - (1-\cos ^2 t)(3\cos t ) \)
\( = \cos ^3 t - 3 \cos t + 3 \cos ^3 t \)
\( = 4 \cos ^3 t - 3 \cos t \)
Q.E.D.