Question

Prove that : ${cos}^{-1}\left(\frac{3}{5}\right)+{cos}^{-1}\left(\frac{4}{5}\right)=\frac{\pi }{2}$

Answer 1

Let,

${\cos }^{-1}\left(\frac{3}{5}\right)=x$

$\Rightarrow \cos x=\frac{3}{5}$

From Pythagorus theorem, we have,

$\sin x=\frac{4}{5}$

Given,

${\cos }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)$

$={\sin }^{-1}\left(\frac{4}{5}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)$

we have general formula, ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$

$=\frac{\pi }{2}$

Hence proved.