The set of values of $\sec ^{-1} \frac{1}{2}$ is ........... .
Since, domain of $\sec ^{-1} x$ is $R-(-1,1)$.
$$\Rightarrow \quad(-\infty,-1] \cup[1, \infty)$$
So, there is no set of values exist for $\sec ^{-1} \frac{1}{2}$.
So, $\phi$ is the answer.
The principal value of $\tan ^{-1} \sqrt{3}$ is ............ .
$\tan ^{-1} \sqrt{3}=\tan ^{-1} \tan \left(\frac{\pi}{3}\right)$
$\left[\because \tan ^{-1}(\tan x)=x, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right]=\left(\frac{\pi}{3}\right)$
The value of $\cos ^{-1}\left(\cos \frac{14 \pi}{3}\right)$ is ............. .
We have, $$\cos ^{-1}\left(\cos \frac{14 \pi}{3}\right)=\cos ^{-1} \cos \left(4 \pi+\frac{2 \pi}{3}\right)$$
$$\begin{array}{lr} =\cos ^{-1} \cos \frac{2 \pi}{3} & {[\because \cos (2 n \pi+\theta)=\cos \theta]} \\ =\frac{2 \pi}{3} & \left\{\because \cos ^{-1}(\cos x)=x, x \in[0, \pi]\right\} \end{array}$$
The value of $\cos \left(\sin ^{-1} x+\cos ^{-1} x\right)$, where $|x| \leq 1$, is ............ .
$$\begin{aligned} &\begin{gathered} \cos \left(\sin ^{-1} x+\cos ^{-1} x\right) \\ =\cos \frac{\pi}{2}=0 \end{gathered}\\ &\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right] \end{aligned}$$
The value of $\tan \left(\frac{\sin ^{-1} x+\cos ^{-1} x}{2}\right)$, when $x=\frac{\sqrt{3}}{2}$, is ............... .
$\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right]$
$\tan \left(\frac{\sin ^{-1} x+\cos ^{-1} x}{2}\right)=\tan \left(\frac{\pi / 2}{2}\right)$
$=\tan \frac{\pi}{4}=1$