First slide

Properties of ITF

Question

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

Moderate

A

$\frac{1}{2}$
B

1
C

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

-1

Solution

$W e h a v e \sin^{- 1} (x - \frac{x^{2}}{2} + \frac{x^{3}}{4} - . . .) + \cos^{- 1} (x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{4} - . . . .) = \frac{π}{2} \Rightarrow x - \frac{x^{2}}{2} + \frac{x^{3}}{4} - . . . = x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{4} - . . . . \Rightarrow x = x^{2} \Rightarrow x (x - 1) = 0 \Rightarrow x = 0, 1 S i n c e 0 < x < \sqrt{2}, \Rightarrow x = 1$

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