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If $\sec x=\frac{25}{24}$ and $x$ lies in first quadrant, then $\sin \frac{x}{2}+\cos \frac{x}{2}=$
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Verified Answer
The correct answer is:
$\frac{8}{5 \sqrt{2}}$
We have $\cos x=\frac{24}{25} \Rightarrow \sin x=\frac{7}{25} \ldots\left[\because x\right.$ lies in $1^{\text {st }}$ quadrant $]$ Also $\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^2=1+\sin x=1+\frac{7}{25}=\frac{32}{25}$
$$
\therefore \sin \frac{x}{2}+\cos \frac{x}{2}=\sqrt{\frac{32}{25}}=\frac{4 \sqrt{2}}{5}=\frac{8}{5 \sqrt{2}}
$$
$$
\therefore \sin \frac{x}{2}+\cos \frac{x}{2}=\sqrt{\frac{32}{25}}=\frac{4 \sqrt{2}}{5}=\frac{8}{5 \sqrt{2}}
$$
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