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Properties of ITF

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Question

${The number of values of x for which sin}^{- 1} (x^{2} - \frac{x^{4}}{3} + \frac{x^{6}}{9} . . . . . .) {+cos}^{- 1} (x^{4} - \frac{x^{8}}{3} + \frac{x^{12}}{9} . . . . . .) = \frac{π}{2},0≤ |x| < \sqrt{3},is$

Moderate

Solution

$\sin^{- 1} (x^{2} - \frac{x^{4}}{3} + \frac{x^{6}}{9} . . . . . .) {+cos}^{- 1} (x^{4} - \frac{x^{8}}{3} + \frac{x^{12}}{9} . . . . . .) = \frac{π}{2}$
$\Rightarrow (x^{2} - \frac{x^{4}}{3} + \frac{x^{6}}{9} . . . . . .) = (x^{4} - \frac{x^{8}}{3} + \frac{x^{12}}{9} . . . . . .)$ $(\sin c e \sin^{- 1} (x) + \cos^{- 1} (x) = \frac{π}{2})$
$\Rightarrow \frac{x^{2}}{1 + \frac{x^{2}}{3}} = \frac{x^{4}}{1 + \frac{x^{4}}{3}}$ $(s u m o f n t e r m s i n G . P = \frac{a}{1 - r})$
$\Rightarrow x^{2} = 0 o r \frac{3}{3 + x^{2}} = \frac{3 x^{2}}{3 + x^{4}}$
$\Rightarrow x^{2} = 1$
$\Rightarrow x = 0, 1 or -1$
$number of values of x is equal to 3$

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