Find the principal values of each of the following:

(i) Let $\sec ^{-1}(-\sqrt{2})=y$

Then,

$\sec y=-\sqrt{2}$

We know that the range of the principal value branch is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$.

Thus,

$\sec y=-\sqrt{2}=\sec \left(\frac{3 \pi}{4}\right)$

$\Rightarrow y=\frac{3 \pi}{4} \in[0, \pi], y \neq \frac{\pi}{2}$

Hence, the principal value of $\sec ^{-1}(-\sqrt{2})$ is $\frac{3 \pi}{4}$.

(iii)

Let $\sec ^{-1}\left(2 \sin \frac{3 \pi}{4}\right)=y$

Then,

$\sec y=2 \sin \frac{3 \pi}{4}$

We know that the range of the principal value branch is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$.

Thus,

$\sec y=2 \sin \frac{3 \pi}{4}=2 \times \frac{1}{\sqrt{2}}=\sqrt{2}=\sec \left(\frac{\pi}{4}\right)$

$\Rightarrow y=\frac{\pi}{4} \in[0, \pi]$

Hence, the principal value of $\sec ^{-1}\left(2 \sin \frac{3 \pi}{4}\right)$ is $\frac{\pi}{4}$.

(iv)

Let $\sec ^{-1}\left(2 \tan \frac{3 \pi}{4}\right)=y$

Then,

$\sec y=2 \tan \frac{3 \pi}{4}$

We know that the range of the principal value branch is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$.

Thus,

$\sec y=2 \tan \frac{3 \pi}{4}=2 \times(-1)=-2=\sec \left(\frac{2 \pi}{3}\right)$

$\Rightarrow y=\frac{2 \pi}{3} \in[0, \pi]$

Hence, the principal value of $\sec ^{-1}\left(2 \tan \frac{3 \pi}{4}\right)$ is $\frac{2 \pi}{3}$.