Write the value of (2 ^ -1 1 3 ). - Vidyadip

Question

Write the value of $\cos\Big(2\sin^{-1}\frac{1}{3}\Big).$

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Answer

Let $\text{y}=\sin^{-1}\frac{1}{3}$
Then, $\sin\text{y}=\frac{1}{3}$
Now, $\cos\text{y}=\sqrt{1-\sin^2\text{y}}$
$\Rightarrow\cos\text{y}=\sqrt{1-\frac{1}{9}}=\sqrt{\frac{8}{9}}=\frac{2\sqrt2}{3}$
$\cos\Big(2\sin^{-1}\frac{1}{3}\Big)=\cos(2\text{y})$
$=\cos^2\text{y}-\sin^2\text{y}$ $\big[\because\ \cos2\text{x}=\cos^2\text{x}-\sin^2\text{x}\big]$
$=\Big(\frac{2\sqrt2}{3}\Big)^2-\Big(\frac{1}{3}\Big)^2$
$=\frac{8}{9}-\frac{1}{9}$
$=\frac{7}{9}$
$\because\ \cos\Big(2\sin^{-1}\frac{1}{3}\Big)=\frac{7}{9}$

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