Prove the following results: ^ -1 4 5 +2 ^ -1 1 3 = 2 - Vidyadip

Question

Prove the following results:
$\sin^{-1}\frac{4}{5}+2\tan^{-1}\frac{1}{3}=\frac{\pi}{2}$

✓

Answer

$\text{L.H.S}=\sin^{-1}\frac{4}{5}+2\tan^{-1}\frac{1}{3}$
$=\sin^{-1}\frac{4}{5}+\tan^{-1}\Bigg\{\frac{2\times\frac{1}{2}}{1-\big(\frac{1}{3}\big)^2}\Bigg\}$ $\Big[\because\ 2\tan^{-1}\text{x}=\tan^{-1}\Big\{\frac{2\text{x}}{1-\text{x}^2}\Big\}\Big]$
$=\sin^{-1}\frac{4}{5}+\tan^{-1}\Bigg\{\frac{\frac{2}{3}}{\frac{8}{9}}\Bigg\}$
$=\sin^{-1}\frac{4}{5}+\tan^{-1}\frac{3}{4}$
$=\sin^{-1}\frac{4}{5}+\cos^{-1}\frac{1}{\sqrt{1+\frac{9}{16}}}$ $\Big[\because\ \tan^{-1}\text{x}=\cos^{-1}\frac{1}{\sqrt{1+\text{x}^2}}\Big]$
$=\sin^{-1}\frac{4}{5}+\cos^{-1}\frac{1}{\frac{5}{4}}$
$=\sin^{-1}\frac{4}{5}+\cos^{-1}\frac{4}{5}$
$=\frac{\pi}{2}=\text{R.H.S}$

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