Question

The value of tan $\left[\cos^{-1}\frac{4}{5}=\sin^{-1}\frac{2}{\sqrt3}\right]$ is

• A. $\frac{7}{16}$
• B. $\frac{17}{6}$
• C. $\frac{6}{17}$
• D. none of these.

Answer 1

Answer: (B)

Let $cos^{-1} \frac{4}{5} = \theta $
$ \therefore cos\,\theta = \frac{4}{5} $

$ \therefore tan\, \theta = \frac{3}{4}$
Let $sin^{-1} \frac{2}{\sqrt{13}} = \phi$
$\Rightarrow sin \,\phi = \frac{2}{\sqrt{13}} $
$ \therefore tan\, \phi = \frac{2}{3} $
$ \therefore tan^{-1} \left[ cos^{-1} \frac{4}{5} +sin^{-1} \frac{2}{\sqrt{13}}\right] $
$ = tan \left( \theta+\phi\right) $
$=\frac{ tan \,\theta +tan\,\phi}{1- tan \,\theta\, tan\, \phi} =\frac{ \frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\cdot\frac{2}{3}} $
$= \frac{\frac{17}{12}}{1-\frac{1}{2}} $
$ = \frac{17}{6}$