The value of $\cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right\}$ is

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Aicharya · Accepted Answer

$\begin{aligned} & \cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left[\cos \left(\frac{9 \pi}{10}ight)-\sin \left(\frac{9 \pi}{10}ight)ight]ight\} \ & =\cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left(\cos \left(\pi-\frac{\pi}{10}ight)-\sin \left(\pi-\frac{\pi}{10}ight)ight)ight\} \ & =\cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left[-\cos \left(\frac{\pi}{10}ight)-\sin \left(\frac{\pi}{10}ight)ight]ight\}\end{aligned}$ $\begin{aligned} & =\cos ^{-1}\left\{(-1)\left[\frac{1}{\sqrt{2}} \cos \left(\frac{\pi}{10}ight)+\frac{1}{\sqrt{2}} \sin \left(\frac{\pi}{10}ight)ight]ight\} \ & =\cos ^{-1}\left\{(-1)\left[\cos \frac{\pi}{4} \cos \frac{\pi}{10}+\sin \frac{\pi}{4} \sin \frac{\pi}{10}ight]ight\}\end{aligned}$ $\begin{aligned} & =\cos ^{-1}\left\{(-1)\left[\cos \left(\frac{\pi}{4}-\frac{\pi}{10}ight)ight]ight\} \ & =\cos ^{-1}\left\{(-1)\left[\cos \left(\frac{3 \pi}{20}ight)ight]ight\} \ & =\pi-\cos ^{-1}\left(\cos \frac{3 \pi}{20}ight) \ & =\pi-\frac{3 \pi}{20} \ & =\frac{17 \pi}{20}\end{aligned}$

The value of $\cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9…

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