The value of tan

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) $\frac{3}{29}$

Solution:

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

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

$\therefore \cos y=\frac{1}{5 \sqrt{2}} \Rightarrow \sin y=\frac{7}{5 \sqrt{2}} \Rightarrow \tan y=7$

$\sin z=\frac{4}{\sqrt{17}} \Rightarrow \cos z=\frac{1}{\sqrt{17}} \Rightarrow \tan z=4$

$\therefore \tan \left(\cos ^{-1} \frac{1}{5 \sqrt{2}}-\sin ^{-1} \frac{4}{\sqrt{17}}\right)=\tan (y-z)$

$=\frac{\tan y-\tan z}{1+\tan y \tan z}$

$=\frac{7-4}{1+7 \times 4}$

$=\frac{3}{29}$