First slide

Properties of ITF

Question

If $\sin^{- 1} x + \sin^{- 1} y = \frac{π}{2}$ , then $\frac{1 + x^{4} + y^{4}}{x^{2} - x^{2} y^{2} + y^{2}}$ is equal to

Moderate

A

1
B

2
C

$\frac{1}{2}$
D

None of these

Solution

$\sin^{- 1} x + \sin^{- 1} y = \frac{π}{2} \Rightarrow \sin^{- 1} x = \frac{π}{2} - \sin^{- 1} y = \cos^{- 1} y$
$\begin{array}{l} \Rightarrow \sin^{- 1} x = \sin^{- 1} \sqrt{1 - y^{2}} \Rightarrow x = \sqrt{1 - y^{2}} \Rightarrow x^{2} + y^{2} = 1 \\ ∴ \frac{1 + x^{4} + y^{4}}{x^{2} - x^{2} y^{2} + y^{2}} = \frac{1 + {(x^{2} + y^{2})}^{2} - 2 x^{2} y^{2}}{1 - x^{2} y^{2}} \end{array}$
$= \frac{1 + 1 - 2 x^{2} y^{2}}{1 - x^{2} y^{2}} = 2$

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