Question

Evaluate :

$\cos \left\{{\tan }^{-1}(-\frac{3}{4})\right\}$

Answer 1

$\cos \left\{{\tan }^{-1}\left(\frac{-3}{4}\right)\right\}$

Let ${\tan }^{-1}\left(\frac{-3}{4}\right)=a$

$\Rightarrow \tan a=\frac{-3}{4}$

$\Rightarrow {\tan }^{2}a=\frac{9}{16}$

$\Rightarrow {\sec }^{2}a=1+{\tan }^{2}a=1+\frac{9}{16}=\frac{16+9}{16}=\frac{25}{16}$

$\Rightarrow \sec a=\frac{5}{4}$

$\Rightarrow \cos a=\frac{1}{\sec a}=\frac{1}{\frac{5}{4}}=\frac{4}{5}$

Now, $\cos \left\{{\tan }^{-1}\left(\frac{-3}{4}\right)\right\}=\cos a=\frac{4}{5}$ $\because ba={\tan }^{-1}\left(\frac{-3}{4}\right)$