First slide

Properties of ITF

Question

The value of $\sin^{2} (\cos^{- 1} \frac{1}{2}) + \cos^{2} (\sin^{- 1} \frac{1}{3}) is$

Moderate

A

17/36
B

59/36
C

36/59
D

none of these

Solution

$\begin{array}{l} \sin^{2} (\cos^{- 1} \frac{1}{2}) + \cos^{2} (\sin^{- 1} \frac{1}{3}) \\ = 1 - \cos^{2} (\cos^{- 1} \frac{1}{2}) + 1 - \sin^{2} (\sin^{- 1} \frac{1}{3}) \\ = 1 - {(\frac{1}{2})}^{2} + 1 - {(\frac{1}{3})}^{2} = 2 - \frac{1}{4} - \frac{1}{9} = \frac{59}{36} \end{array}$

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