First slide

Properties of ITF

Question

Given $0 \leq x \leq \frac{1}{2}$ then the value of $\tan [\sin^{- 1} (\frac{x}{\sqrt{2}} + \frac{\sqrt{1 - x^{2}}}{\sqrt{2}}) - \sin^{- 1} x] is$

Moderate

A

-1
B

1
C

$\frac{1}{\sqrt{3}}$
D

$\sqrt{3}$

Solution

$Put x = \sin θ then \sin^{- 1} (\frac{x}{\sqrt{2}} + \frac{\sqrt{1 - x^{2}}}{\sqrt{2}}) \begin{array}{l} = \sin^{- 1} (\frac{1}{\sqrt{2}} \sin θ + \frac{1}{\sqrt{2}} \cos θ) = \sin^{- 1} \sin (θ + \frac{π}{4}) \\ = θ + \frac{π}{4} \end{array}$
$∴$ given expression $= \tan (θ + \frac{π}{4} - θ) = \tan \frac{π}{4} = 1$

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