First slide

Properties of ITF

Question

$If \cos^{- 1} x + \cos^{- 1} y = \frac{π}{2} and \tan^{- 1} x - \tan^{- 1} y = 0$ , then x² + xy + y², is equal to

Moderate

A

0
B

$\frac{1}{\sqrt{2}}$
C

$\frac{3}{2}$
D

$\frac{1}{8}$

Solution

$∵ \tan^{- 1} x - \tan^{- 1} y = 0 \Rightarrow x = y Also, \cos^{- 1} x + \cos^{- 1} y = \frac{π}{2} \begin{array}{l} \Rightarrow 2 \cos^{- 1} x = \frac{π}{2} \\ \Rightarrow \cos^{- 1} x = \frac{π}{4} \\ \Rightarrow x = \frac{1}{\sqrt{2}} \\ \Rightarrow x^{2} = \frac{1}{2} \\ Hence, x^{2} + xy + y^{2} = 3 x^{2} = \frac{3}{2} \end{array}$

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