First slide

Introduction to ITF

Question

The value of $\sin^{- 1} [\cos {\cos^{- 1} (\cos x) + \sin^{- 1} (\sin x)}]$ where $x \in (\frac{π}{2}, π)$ is equal to

Moderate

A

$\frac{π}{2}$
B

$- π$
C

$π$
D

$- \frac{π}{2}$

Solution

$\sin^{- 1} [\cos {\cos^{- 1} (\cos x) + \sin^{- 1} (\sin x)}]$
$= \sin^{- 1} [\cos (x + π - x)]$ $(∵ \sin^{- 1} (sinx) = \sin^{- 1} (\sin (π - x)) = π - x (∵ x \in (\frac{π}{2}, π) \Rightarrow \frac{π}{2} < x < π \Rightarrow 0 < π - x < \frac{π}{2}))$
$= \sin^{- 1} (\cos π) = \sin^{- 1} (- 1) = - \frac{π}{2}$

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