If $ an x=\frac{3}{4}$ and $\pi \lt x \lt \frac{3 \pi}{2}$, then $\cos \frac{x}{2}=$ $\qquad$

Question

If $	an x=\frac{3}{4}$ and $\pi \lt x \lt \frac{3 \pi}{2}$, then $\cos \frac{x}{2}=$ $\qquad$

Aicharya · Accepted Answer

$\begin{array}{ll} & 	an x=\frac{3}{4}, \pi \lt x \lt \frac{3 \pi}{2} \ 	herefore \quad & 1+	an ^2 x=\sec ^2 x \ 	herefore \quad & \sec ^2 x=\frac{25}{16} \ 	herefore \quad & \sec x=\frac{-5}{4} \quad \ldots\left[\because \pi \lt x \lt \frac{3 \pi}{2}ight]\end{array}$ $	herefore \quad \cos x=\frac{-4}{5}$ $	herefore \quad \cos \frac{x}{2}=-\sqrt{\frac{1+\cos x}{2}}$ $\ldots\left[\pi \lt x \lt \frac{3 \pi}{2} \Rightarrow \frac{\pi}{2} \lt \frac{x}{2} \lt \frac{3 \pi}{4}ight]$ $\begin{aligned} & =-\sqrt{\frac{1}{10}} \ & =\frac{-1}{\sqrt{10}}\end{aligned}$

If $\tan x=\frac{3}{4}$ and $\pi \lt x \lt \frac{3 \pi}{2}$, then $\cos \frac{x}{2}=$ $\qquad$

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