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Please solve RD Sharma class 12 Chapter Inverse Trigonometric Functions exercise 3.5 question 1 sub question (ii) maths textbook solution.

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Hints: Domain of \operatorname{cosec}^{-1} s R- (-1,1) and the range of \operatorname{cosec}^{-1}  is \left [ \frac{-\pi}{2}, \frac{\pi}{2} \right ] \left \{ 0 \right \}

Given: \operatorname{cosec}^{-1}(-\sqrt{2})

Explanation: Let y=\operatorname{cosec}^{-1}(-\sqrt{2})

\operatorname{cosec} y=-2

\operatorname{cosec} y =- \operatorname{cosec}\frac{\pi}{6}                                                                                                 \left [\operatorname{cosec}\frac{-\pi}{6}=2 \right ]

\therefore y =\frac{-\pi}{6}

Since, range of \operatorname{cosec}^{-1} is \left [ \frac{-\pi}{2}, \frac{\pi}{2} \right ] \left \{ 0 \right \}

Therefore, principal value is \frac{-\pi}{6}

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