Q65 / 160 · TS EAMCET 2020 (09 Sep Shift 1) · Mathematics

$\sin h^{- 1} (- 2) + cosec h^{- 1} (- 2) + \cot h^{- 1} (- 2) =$

Solution

We know that $\sin h^{- 1} (x) = \log (x + \sqrt{x^{2} + 1}), cosec h^{- 1} (x) = \log (\frac{1}{x} + \sqrt{\frac{1}{x^{2}} + 1})$

$\cot h^{- 1} (x) = \frac{1}{2} \log (\frac{x + 1}{x - 1})$

Now

$\sinh^{- 1} (- 2) + {cosech}^{- 1} (- 2) + {coth}^{- 1} (- 2) =$

$= \log [(- 2 + \sqrt{5}) (\frac{- 1 + \sqrt{5}}{2}) (\sqrt{\frac{1}{3}})] = \log (\frac{7 - 3 \sqrt{5}}{2 \sqrt{3}})$

Hence, the correct answer is option: 'A'.