Question

The value of tan $\left[{\cos }^{-1}\frac{4}{5}={\sin }^{-1}\frac{2}{\sqrt{3}}\right]$ is

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

Answer 1

Answer: (B)

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

$\therefore tan\theta =\frac{3}{4}$
Let $si{n}^{-1}\frac{2}{\sqrt{13}}=\varphi$
$\Rightarrow sin\varphi =\frac{2}{\sqrt{13}}$
$\therefore tan\varphi =\frac{2}{3}$
$\therefore ta{n}^{-1}\left[co{s}^{-1}\frac{4}{5}+si{n}^{-1}\frac{2}{\sqrt{13}}\right]$
$=tan\left(\theta +\varphi \right)$
$=\frac{tan\theta +tan\varphi }{1-tan\theta tan\varphi }=\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\cdot \frac{2}{3}}$
$=\frac{\frac{17}{12}}{1-\frac{1}{2}}$
$=\frac{17}{6}$