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Question: If \(\cos\alpha\cosh\beta = 1,\) then \(\beta\) is equal to...

If cos⁡αcosh⁡β=1,\cos\alpha\cosh\beta = 1, then β\beta is equal to

A

logsec(α2){logsec}\left( \frac{\alpha}{2} \right)

B

logtanα{logtan}\alpha

C

log⁡(sec⁡α+tan⁡α)\log(\sec\alpha + \tan\alpha)

D

logsin(α2){logsin}\left( \frac{\alpha}{2} \right)

Answer

log⁡(sec⁡α+tan⁡α)\log(\sec\alpha + \tan\alpha)

Explanation

Solution

cos⁡α.cosh⁡β=1\cos\alpha.\cosh\beta = 1 ⇒ cosh⁡β=sec⁡α\cosh\beta = \sec\alpha⇒β=cos⁡h−1(sec⁡α)\beta = {\cos h}^{- 1}(\sec\alpha) =

log⁡(sec⁡α+sec⁡2α−1)\log(\sec\alpha + \sqrt{\sec^{2}\alpha - 1}) = log⁡(sec⁡α+tan⁡α)\log(\sec\alpha + \tan\alpha)