First slide

Properties of ITF

Question

If $0 < x < 1$ then $T a n^{- 1} (\frac{\sqrt{1 - x^{2}}}{1 + x}) =$

Moderate

A

$\frac{1}{2} \cos^{- 1} x$
B

$\cos^{- 1} \sqrt{\frac{1 + x}{2}}$
C

$\sin^{- 1} \sqrt{\frac{1 - x}{2}}$
D

All the above

Solution

$G i v e n 0 < x < 1 . L e t x = \cos 2 θ . T h e n T a n^{- 1} (\frac{\sqrt{1 - x^{2}}}{1 + x}) = T a n^{- 1} (\frac{\sqrt{(1 - x) (1 + x)}}{1 + x}) = T a n^{- 1} (\sqrt{\frac{1 - x}{1 + x}}) = T a n^{- 1} \sqrt{\frac{1 - \cos 2 θ}{1 + \cos 2 θ}} = T a n^{- 1} \sqrt{\frac{2 \sin^{2} θ}{2 \cos^{2} θ}} = T a n^{- 1} (\tan θ) = θ = \frac{1}{2} \cos^{- 1} x (∵ x = \cos 2 θ) A l s o x = \cos 2 θ = 2 \cos^{2} θ - 1 \Rightarrow \cos^{2} θ = \frac{1 + x}{2} \Rightarrow \cos θ = \sqrt{\frac{1 + x}{2}} \Rightarrow θ = \cos^{- 1} \sqrt{\frac{1 + x}{2}} A l s o x = \cos 2 θ = 1 - 2 \sin^{2} θ \Rightarrow \sin^{2} θ = \frac{1 - x}{2} \Rightarrow \sin θ = \sqrt{\frac{1 - x}{2}} \Rightarrow θ = \sin^{- 1} \sqrt{\frac{1 - x}{2}}$

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