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If $\sin \mathrm{x}=\frac{\sqrt{5}}{3}$ and $0<\mathrm{x}<\frac{\pi}{2}$ find the values of $\cos 2 x$



Given: $\sin \mathrm{x}=\frac{\sqrt{5}}{3}$

To find: $\cos 2 x$

We know that

$\cos 2 x=1-2 \sin ^{2} x$

Putting the value, we get

$\cos 2 x=1-2\left(\frac{\sqrt{5}}{3}\right)^{2}$

$\cos 2 x=1-2 \times \frac{5}{9}$

$\cos 2 x=1-\frac{10}{9}$

$\cos 2 x=\frac{9-10}{9}$

$\therefore \cos 2 x=-\frac{1}{9}$


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