First slide

Properties of ITF

Question

If $S i n^{- 1} (x - \frac{x^{2}}{2} + \frac{x^{3}}{4} - ...) + C o s^{- 1} (x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{4} - ...) = \frac{π}{2}$ for $0 < | x | < \sqrt{2}$ then x=

Easy

A

$\frac{1}{2}$
B

1
C

$- \frac{1}{2}$
D

$- 1$

Solution

$W e h a v e \sin^{- 1} (x - \frac{x^{2}}{2} + \frac{x^{3}}{4} - . . . . .) + \cos^{- 1} (x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{4} - . . . . .) = \frac{π}{2} \Rightarrow x - \frac{x^{2}}{2} + \frac{x^{3}}{4} - . . . . . = x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{4} - . . . . . \Rightarrow \frac{x}{1 + \frac{x}{2}} = \frac{x^{2}}{1 + \frac{x^{2}}{2}} \Rightarrow \frac{1}{2 + x} = \frac{x}{2 + x^{2}} \Rightarrow 2 + x^{2} = 2 x + x^{2} \Rightarrow x = 1$

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