First slide

Properties of ITF

Question

$2 \tan^{- 1} [\sqrt{\frac{a - b}{a + b}} \tan \frac{θ}{2}] =$

Moderate

A

$\cos^{- 1} (\frac{a \cos θ + b}{a + b \cos θ})$
B

$\cos^{- 1} (\frac{a + b \cos θ}{a \cos θ + b})$
C

$\cos^{- 1} (\frac{a \cos θ}{a + b \cos θ})$
D

$\cos^{- 1} (\frac{b \cos θ}{a \cos θ + b})$

Solution

$2 \tan^{- 1} [\sqrt{\frac{a - b}{a + b}} \tan \frac{θ}{2}] =$ $\cos^{- 1} [\frac{1 - (\frac{a - b}{a + b}) \tan^{2} \frac{θ}{2}}{1 + (\frac{a - b}{a + b}) \tan^{2} \frac{θ}{2}}]$
$(∵ 2 \tan^{- 1} x = \cos^{- 1} \frac{1 - x^{2}}{1 + x^{2}})$
$\begin{array}{l} = \cos^{- 1} [\frac{(a + b) - (a - b) \tan^{2} \frac{θ}{2}}{(a + b) + (a - b) \tan^{2} \frac{θ}{2}}] \\ = \cos^{- 1} [\frac{a (1 - \tan^{2} \frac{θ}{2}) + b (1 + \tan^{2} \frac{θ}{2})}{a (1 + \tan^{2} \frac{θ}{2}) + b (1 - \tan^{2} \frac{θ}{2})}] \\ = \cos^{- 1} [\frac{\frac{a (1 - \tan^{2} \frac{θ}{2})}{1 + \tan^{2} \frac{θ}{2}} + b}{a + b (\frac{1 - \tan^{2} \frac{θ}{2}}{1 + \tan^{2} \frac{θ}{2}})}] \\ = \cos^{- 1} [\frac{a \cos θ + b}{a + b \cos θ}] \end{array}$

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