Question

Evaluate: $\tan \left[{\cos }^{-1}\left(\frac{4}{5}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right]$

• A. $\left(\frac{6}{17}\right)$
• B. $\left(\frac{17}{6}\right)$
• C. $\left(\frac{7}{16}\right)$
• D. $\left(\frac{16}{7}\right)$

Answer 1

The correct option is B

$\left(\frac{17}{6}\right)$

Explanation for correct option

Evaluating the given trigonometric expression

$\tan \left[{\cos }^{-1}\left(\frac{4}{5}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right]=\tan \left[{\tan }^{-1}\left(\frac{\sqrt{1-{\left({\displaystyle \frac{4}{5}}\right)}^2}}{{\displaystyle \frac{4}{5}}}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right]\mathbf{}\mathbf{[}\mathbf{\text{∵}}\mathbf{}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\frac{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}{\mathbf{x}}\mathbf{\right)}\mathbf{]}$

$$\begin{gathered} =\tan \left[{\tan }^{-1}\left(\frac{\sqrt{1-{\displaystyle \frac{16}{25}}}}{{\displaystyle \frac{4}{5}}}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right] \\ =\tan \left[{\tan }^{-1}\left(\frac{\sqrt{{\displaystyle \frac{9}{25}}}}{{\displaystyle \frac{4}{5}}}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right] \\ =\tan \left[{\tan }^{-1}\left(\frac{{\displaystyle \frac{3}{5}}}{{\displaystyle \frac{4}{5}}}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right] \\ =\tan \left[{\tan }^{-1}\left(\frac{3}{4}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right] \\ =\tan \left[{\tan }^{-1}\left(\frac{{\displaystyle \frac{3}{4}+\frac{2}{3}}}{1-{\displaystyle \frac{3}{4}}\times {\displaystyle \frac{2}{3}}}\right)\right][\because {\tan }^{-1}\left(\mathrm{x}\right)+{\tan }^{-1}\left(\mathrm{y}\right)={\tan }^{-1}\left(\frac{\mathrm{x}+\hspace{0.17em}\mathrm{y}}{1-\mathrm{xy}}\right)] \\ =\tan \left[{\tan }^{-1}\left(\frac{{\displaystyle \frac{9+8}{12}}}{{\displaystyle \frac{12-6}{12}}}\right)\right] \\ =\tan \left[{\tan }^{-1}\left(\frac{17}{6}\right)\right] \\ =\frac{17}{6} \end{gathered}$$

Hence, the correct answer is Option (B).