Question
$${\text{cose}}{{\text{c}}^{ - 1}}\left( {\cos x} \right)$$ is real if
A.
$$x \in \left[ { - 1,1} \right]$$
B.
$$x \in R$$
C.
$$x$$ is an odd multiple of $$\frac{\pi }{2}$$
D.
$$x$$ is a multiple of $$\pi $$
Answer :
$$x$$ is a multiple of $$\pi $$
Solution :
$${\text{cose}}{{\text{c}}^{ - 1}}\left( {\cos x} \right)$$ exists if $$\cos x \leqslant - 1$$ or $$\cos x \geqslant 1.$$ But $$ - 1 \leqslant \cos x \leqslant 1.$$
$$\therefore {\text{cose}}{{\text{c}}^{ - 1}}\left( {\cos x} \right)$$ exists if $$\cos x = 1\,{\text{or,}} - 1$$
$$ \Rightarrow \,\,x = n\pi ,n \in {\Bbb Z}.$$