find the exact value of the expression, if it is defined. (if an answer is undefined, enter…

Explanation:

Step1: Evaluate inner - inverse sine

Let $\theta=\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$. By the definition of the inverse - sine function, $\sin\theta=\frac{\sqrt{2}}{2}$ and $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$. The value of $\theta$ in this range for which $\sin\theta=\frac{\sqrt{2}}{2}$ is $\theta = \frac{\pi}{4}$.

Step2: Evaluate outer - tangent

Now we need to find $\tan\theta$ where $\theta=\frac{\pi}{4}$. Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, and for $\theta=\frac{\pi}{4}$, $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, then $\tan\frac{\pi}{4}=\frac{\sin\frac{\pi}{4}}{\cos\frac{\pi}{4}} = 1$.

Answer:

$1$