91Ó°ÊÓ

Find the principal values of the following: $$ \cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right) $$

Find the principal values of the following: $$ \operatorname{cosec}^{-1}(\sqrt{2}) $$

Write the following functions in the simplest form: $$ \tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a>0 ; \frac{-a}{\sqrt{3}}

Find the values of the following: $$ \tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right) $$

Find the values of each of the following: $$ \tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right] $$

Find the values of the following: $$ \cos ^{-1}\left(\frac{1}{2}\right)+2 \sin ^{-1}\left(\frac{1}{2}\right) $$

Find the values of each of the following: $$ \tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right],|x|<1, y>0 \text { and } x y<1 $$

Find the values of the following: \(\tan ^{-1} \sqrt{3}-\sec ^{-1}(-2)\) is equal to (A) \(\pi\) (B) \(-\frac{\pi}{3}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{2 \pi}{3}\)

Find the values of each of the following: $$ \text { If } \tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4}, \text { then find the value of } x $$

Solve the following equations: \(\sin \left(\tan ^{-1} x\right),|x|<1\) is equal to (A) \(\frac{x}{\sqrt{1-x^{2}}}\) (B) \(\frac{1}{\sqrt{1-x^{2}}}\) (C) \(\frac{1}{\sqrt{1+x^{2}}}\) (D) \(\frac{x}{\sqrt{1+x^{2}}}\)