Answer
(a)
$$
\begin{aligned}
\frac{d}{d x}\left(\sin ^{n} x \cos n x\right) &=n \sin ^{n-1} x \cos x \cos n x+\sin ^{n} x(-n \sin n x) \\
& \,\,\,\,\,\,\,\,\,\,\ \text{ [Product Rule] }\\
&=n \sin ^{n-1} x(\cos n x \cos x-\sin n x \sin x) \\
& \,\,\,\,\,\,\,\,\,\,\ \text{ [Addition Formula for cosine]}\\
&=n \sin ^{n-1} x \cos (n x+x) \\
&=n \sin ^{n-1} x \cos [(n+1) x]
\end{aligned}
$$
(b)
$$
\begin{aligned}
\frac{d}{d x}\left(\cos ^{n} x \cos n x\right) &=n \cos ^{n-1} x(-\sin x) \cos n x+\cos ^{n} x(-n \sin n x) \\
& \quad \quad \quad \text{ [ Product Rule ]}\\
&=-n \cos ^{n-1} x(\cos n x \sin x+\sin n x \cos x) \\
& \quad \quad \quad \text{ [ Addition Formula for sine ]}\\
& =-n \cos ^{n-1} x \sin (n x+x)\\
& =-n \cos ^{n-1} x \sin [(n+1) x]
\end{aligned}
$$
Work Step by Step
(a)
$$
\begin{aligned}
\frac{d}{d x}\left(\sin ^{n} x \cos n x\right) &=n \sin ^{n-1} x \cos x \cos n x+\sin ^{n} x(-n \sin n x) \\
& \,\,\,\,\,\,\,\,\,\,\ \text{ [Product Rule] }\\
&=n \sin ^{n-1} x(\cos n x \cos x-\sin n x \sin x) \\
& \,\,\,\,\,\,\,\,\,\,\ \text{ [Addition Formula for cosine]}\\
&=n \sin ^{n-1} x \cos (n x+x) \\
&=n \sin ^{n-1} x \cos [(n+1) x]
\end{aligned}
$$
(b)
$$
\begin{aligned}
\frac{d}{d x}\left(\cos ^{n} x \cos n x\right) &=n \cos ^{n-1} x(-\sin x) \cos n x+\cos ^{n} x(-n \sin n x) \\
& \quad \quad \quad \text{ [ Product Rule ]}\\
&=-n \cos ^{n-1} x(\cos n x \sin x+\sin n x \cos x) \\
& \quad \quad \quad \text{ [ Addition Formula for sine ]}\\
& =-n \cos ^{n-1} x \sin (n x+x)\\
& =-n \cos ^{n-1} x \sin [(n+1) x]
\end{aligned}
$$
