Answer
$\tan{\theta}\sin{\theta}+\cos{\theta}=\sec{\theta}$
Work Step by Step
Use a graphing utility to graph $y=\tan{\theta}\sin{\theta}+\cos{\theta}$.
Refer to the graph below.
Notice that the graph looks the same as that of $y=\sec{\theta}$.
RECALL:
(1) $\tan{\theta}=\dfrac{\sin{\theta}}{\cos{\theta}}\\\\$
(2) $\sec{\theta}=\dfrac{1}{\cos{\theta}}$
Use the definitions above to obtain:
\begin{align*}
\tan{\theta}\sin{\theta}+\cos{\theta}&=\frac{\sin{\theta}}{\cos{\theta}}\cdot \sin{\theta}+\cos{\theta}\\\\
&=\frac{\sin^2{\theta}}{\cos{\theta}}+\cos{\theta}\\\\
&=\frac{\sin^2{\theta}}{\cos{\theta}}+\frac{\cos^2{\theta}}{\cos{\theta}}\\\\
&=\frac{\sin^2{\theta}+\cos^2{\theta}}{\cos{\theta}}\\\\
&=\frac{1}{\cos{\theta}}\\\\
&=\sec{\theta}
\end{align*}
Therefore,
$$\tan{\theta}\sin{\theta}+\cos{\theta}=\sec{\theta}$$
