Question

If we consider only the principal values of the inverse trigonometric functions, then the values of $\tan [{\cos }^{-1}\frac{1}{\sqrt{2}}-{\sin }^{-1}\frac{4}{\sqrt{17}}]$ is

• A. $\frac{\sqrt{29}}{3}$
• B. $\frac{29}{3}$
• C. $\frac{\sqrt{3}}{29}$
• D. $-\frac{3}{5}$

Answer 1

Answer: (D)

$-\frac{3}{5}$

${\cos }^{-1}\frac{1}{\sqrt{2}}={\tan }^{-1}1,{\sin }^{-1}\frac{4}{\sqrt{17}}={\tan }^{-1}4$

Therefore $\tan [{\cos }^{-1}\frac{1}{\sqrt{2}}-{\sin }^{-1}\frac{4}{\sqrt{17}}]=\tan [{\tan }^{-1}1-{\tan }^{-1}4]$

$=\tan \left[{\tan }^{-1}\left(\frac{1-4}{1+4}\right)\right]=\tan \left[{\tan }^{-1}(-\frac{3}{5})\right]=-\frac{3}{5}$