First slide

Properties of ITF

Question

$If x \in (7 π, 8 π), then \tan^{- 1} \sqrt{\frac{1 - \cos x}{1 + \cos x}} =$

Easy

A

$- \frac{x}{2}$
B

$\frac{x}{2}$
C

$4 π - \frac{x}{2}$
D

none of these

Solution

$\tan^{- 1} \sqrt{\frac{1 - \cos x}{1 + \cos x}} = \tan^{- 1} (- \tan \frac{x}{2}) \begin{array}{l} = \tan^{- 1} \tan (4 π - \frac{x}{2}) = 4 π - \frac{x}{2} \\ (∵ 7 π < x < 8 π \Rightarrow \frac{7 π}{2} < \frac{x}{2} < 4 π, sotan \frac{x}{2} < 0) \end{array}$

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