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Find the Exact value of the expression $$\sin ^{-1}(\sin (7 \pi / 3))$$

$$\sin ^{-1}(\sin (7 \pi / 3))=$$

$$\sin ^{-1}\left(\sin \left(2 \pi+\frac{\pi}{3}\right)\right)$$

$$=\sin ^{-1}\left(\sin \left(\frac{\pi}{3}\right)\right)$$

$$=\frac{\pi}{3}$$

Find the value of $$: \cos ^{-1}(\sin (5 \pi / 4))$$

$$\cos ^{-1}\left(\sin \left(\frac{5 \pi}{4}\right)\right)=\cos ^{-1}\left(\sin \left(\frac{4 \pi}{4}+\frac{\pi}{4}\right)\right)$$

$$=\cos ^{-1}\left(\sin \left(\pi+\frac{\pi}{4}\right)\right)$$

$$\sin (\pi+\theta)=-\sin \theta$$

$$=\cos ^{-1}\left(-\sin \frac{\pi}{4}\right)$$

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

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

$$\pi-\cos ^{-1}(\theta)=\cos ^{-1}(-\theta)$$

$$=\pi-\frac{\pi}{4}=\frac{4 \pi}{4}-\frac{\pi}{4}=\frac{3 \pi}{4}$$

Find the Exact value of the expression : $$\tan \left(\sec ^{-1} 4\right)$$

$$=\frac{4}{1}=4$$

$$\sec^{-1} \sec  \theta=\sec^{-1} 4$$

$$\theta=\sec ^{-1}(4)$$

$$\tan \left(\sec ^{-1}(4)\right)=\tan (\theta)$$

$$=\frac{\sqrt{15}}{1}$$

$$\tan \theta=\sqrt{15}$$

Find the exact value of : $$\tan \left(\sec ^{-1}(-5 / 3)\right)$$

$$\tan \left(\sec ^{-1}\left(\frac{-5}{3}\right)\right)=\tan \left(\pi+\sec ^{-1}\left(\frac{5}{3}\right)\right)$$

$$\sec ^{-1}(-\theta)=\sec ^{-1}(\theta)$$

$$=\tan \left(\sec ^{-1}\left(\frac{5}{3}\right)\right)$$

$$=\frac{5}{3}$$

$$=\frac{4}{3}$$

$$\theta=\sec^{-1} \left(\frac{5}{3}\right)$$

$$\tan \left(\sec ^{-1}\left(\frac{5}{3}\right)\right)=\tan (\theta)=\frac{4}{3}$$

$$\tan \left(\sec ^{-1}\left(\frac{-5}{3}\right)\right)=\frac{4}{3}$$

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