First slide

Properties of ITF

Question

$If {Sin}^{- 1} (x^{2} - \frac{x^{4}}{3} + \frac{x^{6}}{9} - \dots \dots . .) + \cos^{- 1} (x^{4} - \frac{x^{8}}{3} + \frac{x^{12}}{9} - \dots .) = \frac{π}{2} where O \leq x < \sqrt{3}, then the number$ of values of x is

Moderate

A

0
B

1
C

2
D

3

Solution

We have
$\frac{x^{2}}{1 + \frac{x^{2}}{3}} = \frac{x^{4}}{1 + \frac{x^{4}}{3}} (\sin^{- 1} x + \cos^{- 1} x = \frac{π}{2})$
$\frac{3 x^{2}}{3 + x^{2}} = \frac{3 x^{4}}{3 + x^{4}}$
$x^{2} (3 + x^{4}) = x^{4} (3 + x^{2})$
$x^{2} [3 + x^{4} - 3 x^{2} - x^{4}] = 0$
$x = 0, x^{2} = 1$
$x = \pm 1$

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