Question

The value of tan $\left\{{\cos }^{-1}\frac{1}{5\sqrt{2}}-{\sin }^{-1}\frac{4}{\sqrt{17}}\right\}$ is
(a) $\frac{\sqrt{29}}{3}$

(b) $\frac{29}{3}$

(c) $\frac{\sqrt{3}}{29}$

(d) $\frac{3}{29}$

Answer 1

(d) $\frac{3}{29}$

$\mathrm{Let},{\cos }^{-1}\frac{1}{5\sqrt{2}}=y\mathrm{and}{\sin }^{-1}\frac{4}{\sqrt{17}}=z$

$$\begin{gathered} \therefore \cos y=\frac{1}{5\sqrt{2}}\Rightarrow \sin y=\frac{7}{5\sqrt{2}}\Rightarrow \tan y=7 \\ \sin z=\frac{4}{\sqrt{17}}\Rightarrow \cos z=\frac{1}{\sqrt{17}}\Rightarrow \tan z=4 \end{gathered}$$

$$\begin{gathered} \therefore \tan \left({\cos }^{-1}\frac{1}{5\sqrt{2}}-{\sin }^{-1}\frac{4}{\sqrt{17}}\right)=\tan \left(y-z\right) \\ =\frac{\tan y-\tan z}{1+\tan y\tan z} \\ =\frac{7-4}{1+7\times 4} \\ =\frac{3}{29} \end{gathered}$$