Prove the following results: 2 ^ -1 3 5 = ^ -1 24 7 - Vidyadip

Question

Prove the following results:
$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

✓

Answer

$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

$\text{L.H.S}=2\sin^{-1}\frac{3}{5}$

$=2\times\tan^{-1}\begin{bmatrix}\frac{\frac{3}{5}}{\sqrt{1-\Big(\frac{3}{5}\Big)^2}}\end{bmatrix}$ $\bigg\{\text{Since }\sin^{-1}\text{x}=\tan^{-1}\frac{\text{x}}{\sqrt{1-\text{x}^2}}\bigg\}$

$=2\tan^{-1}\Bigg(\frac{\frac{3}{5}}{\frac{4}{5}}\Bigg)$

$=2\tan^{-1}\Big(\frac{3}{4}\Big)$

$=\tan^{-1}\Bigg(\frac{2\times\frac{3}{4}}{1-\big(\frac{3}{4}\big)^2}\Bigg)$ $\Big\{\text{Since }2\tan^{-1}\text{x}=\tan^{-1}\frac{2\text{x}}{1-\text{x}^2}\Big\}$

$=\tan^{-1}\bigg(\frac{\frac{3}{2}}{\frac{7}{16}}\bigg)$

$=\tan^{-1}\Big(\frac{24}{7}\Big)$

$=\text{R.H.S}$

So,

$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

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