Evaluate (1 2 ^ -1 4 5 ). - Vidyadip

Question

Evaluate $\sin\Big(\frac{1}{2}\sin^{-1}\frac{4}{5}\Big).$

✓

Answer

$\sin\Big(\frac{1}{2}\sin^{-1}\frac{4}{5}\Big).$
$=\sin\Bigg(\frac{1}{2}\times2\tan^{-1}\sqrt{\frac{1-\frac{4}{5}}{1+\frac{4}{5}}}\Bigg)$ $\Big\{\text{Since},\cos^{-1}\text{x}=2\tan^{-1}\sqrt{\frac{1-\text{x}}{1+\text{x}}}\Big\}$
$=\sin\Big(\tan^{-1}\frac{1}{3}\Big)$
$=\sin\begin{pmatrix}\sin^{-1}\frac{\frac{1}{3}}{\sqrt{1+\big(\frac{1}{3}\big)^2}}\end{pmatrix}$ $\Big\{\text{Since},\tan^{-1}\text{x}=\sin^{-1}\frac{\text{x}}{\sqrt{1+\text{x}^2}}\Big\}$
$=\frac{\frac{1}{3}}{\frac{\sqrt{10}}{3}}$
$=\frac{1}{\sqrt{10}}$
$\sin\Big(\frac{1}{2}\cos^{-1}\frac{4}{5}\Big)=\frac{1}{\sqrt{10}}$

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