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# The principal value of $\sin^{-1}\tan\big(\large\frac{-5\pi}{4}\big)$ is

$(a)\;\large\frac{\pi}{4}$$\qquad(b)\;-\large\frac{\pi}{4}$$\qquad(c)\;\large\frac{\pi}{2}$$\qquad(d)\;\large\frac{-\pi}{2} Can you answer this question? ## 1 Answer 0 votes \sin^{-1}\{\tan\big(\large\frac{-5\pi}{4})\} \Rightarrow \sin^{-1}\{-\tan(\pi+\large\frac{\pi}{4})\} \Rightarrow \sin^{-1}\big(-\tan\large\frac{\pi}{4}\big) We know that \tan\large\frac{\pi}{4}$$=1$
Hence $\sin^{-1}\big(-1\big)=-\sin^{-1}(1)$
$\qquad\qquad\qquad\quad\;=-\sin^{-1}\big(\sin\large\frac{\pi}{2}\big)$
$\qquad\qquad\qquad\quad\;=-\large\frac{\pi}{2}$
Hence (d) is the correct answer.