Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 53

Determine the equation of the tangent line to the curve $y= \sin (\sin x)$
at the point $(\pi, 0)$.

Solving for the slope


$
\begin{equation}
\begin{aligned}

y' = m =& \frac{d}{dx} [\sin (\sin x)]
\\
\\
m =& \cos (\sin x) \cdot \frac{d}{dx} ( \sin x)
\\
\\
m =& \cos (\sin x ) \cos x
\\
\\
m =& [\cos (\sin \pi)] (\cos \pi)
\\
\\
m =& (1)(-1)
\\
\\
m =& - 1


\end{aligned}
\end{equation}
$



Using the Point Slope Form



$
\begin{equation}
\begin{aligned}

y - y_1 =& m (x - x_1)
\\
\\
y - 0 =& -1 (x - \pi)
\\
\\
y =& -x + \pi
\qquad \qquad \text{Equation of the tangent line at $(\pi,0)$}

\end{aligned}
\end{equation}
$