First slide

Properties of ITF

Question

The value of $\cot^{- 1} [\frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}]$ is

Moderate

A

$π - x$
B

$π - \frac{x}{2}$
C

$π - \frac{x}{3}$
D

None of these

Solution

$\begin{array}{l} \cot^{- 1} [\frac{(\cos \frac{1}{2} x - \sin \frac{1}{2} x) + (\cos \frac{1}{2} x + \sin \frac{1}{2} x)}{(\cos \frac{1}{2} x - \sin \frac{1}{2} x) - (\cos \frac{1}{2} x + \sin \frac{1}{2} x)}] \\ = \cot^{- 1} (- \cot \frac{1}{2} x) \\ = \cot^{- 1} \cot (π - \frac{1}{2} x) = π - \frac{1}{2} x \end{array}$

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