Evaluate exactly.
-$\csc \left( \tan ^ { - 1 } 1 + \csc ^ { - 1 } 1 \right)$
A) $\frac { \sqrt { 2 } } { 2 }$
B) $- \sqrt { 2 }$
C) $\sqrt { 2 }$
D) 0

Question

Quizplus · Accepted Answer

We begin by using the identity $\csc^{-1}x=\sin^{-1}\left(\frac{1}{x}\right)$, so we have:

$$\csc\left(\tan^{-1}1+\csc^{-1}1\right)=\csc\left(\left(\frac{\pi}{4}\right)+\sin^{-1}1\right)$$

Now, we know that $\sin^{-1}1=\frac{\pi}{2}$, so we have:

\begin{align*}
\csc\left(\left(\frac{\pi}{4}\right)+\sin^{-1}1\right)&=\csc\left(\frac{\pi}{4}+\frac{\pi}{2}\right)\
&=\csc\frac{3\pi}{4}\
&=-\sqrt{2}
\end{align*}

Therefore, the answer is $\boxed{\textbf{(C)}\ \sqrt{2}}$.

Evaluate Exactly csc⁡(tan⁡−11+csc⁡−11)\csc \left( \tan ^ { - 1 } 1 + \csc ^ { - 1 } 1 \right)csc(tan−11+csc−11)

Evaluate Exactly $\csc \left( \tan ^ { - 1 } 1 + \csc ^ { - 1 } 1 \right)$