Answer
$-\sin x \sec^{2}(\cos x)$
Work Step by Step
$y= \tan(\cos x)$
Let $\cos x= t$
Then, $y= \tan t$.
According to the chain rule,
$\frac{dy}{dx}= \frac{dy}{dt}\cdot\frac{dt}{dx}$
Therefore, $\frac{dy}{dx}=\sec^{2}t\times-\sin x$
$=-\sin x \sec^{2}(\cos x)$
