The correct option is A log_e(3(2+√(3))) sech^ 1(x)=ln(1+√(1 x^2)x) Similarly cosech^ 1(x)=ln(1+√(x^2+1)x) Hence #160; sech^ 1(12) =ln( ...

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sech-112-cose...

Question

$s e c h^{- 1} (\frac{1}{2}) - c o s e c h^{- 1} (\frac{3}{4}) =$

l o g_{e} (3 (2 + \sqrt{3}))

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l o g_{e} (\frac{1 + \sqrt{3}}{3})

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l o g_{e} (\frac{2 + \sqrt{3}}{3})

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l o g_{e} (\frac{2 - \sqrt{3}}{3})

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Solution

The correct option is A $l o g_{e} (3 (2 + \sqrt{3}))$
$s e c h^{- 1} (x) = l n (\frac{1 + \sqrt{1 - x^{2}}}{x})$
Similarly
$c o s e c h^{- 1} (x) = l n (\frac{1 + \sqrt{x^{2} + 1}}{x})$

Hence
$s e c h^{- 1} (\frac{1}{2})$

$= l n (2 + \sqrt{3})$

and,
$c o s e c h^{- 1} (\frac{3}{4})$

$= l n (\frac{4 + 5}{3})$

$= l n 3$

Hence

$s e c h^{- 1} (\frac{1}{2}) + c o s e c h^{- 1} (\frac{3}{4})$

$= l n (2 + \sqrt{3}) + l n 3$

$= l n (3 (2 + \sqrt{3}))$

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