First slide

Properties of ITF

Question

The value of $\sin^{- 1} (\frac{12}{13}) - \sin^{- 1} (\frac{3}{5})$ is equal to

Moderate

A

$π - \sin^{- 1} (\frac{63}{65})$
B

$\frac{π}{2} - \sin^{- 1} (\frac{56}{65})$
C

$\frac{π}{2} - \cos^{- 1} (\frac{9}{65})$
D

$π - \cos^{- 1} (\frac{33}{65})$

Solution

$We have, \sin^{- 1} (\frac{12}{13}) - \sin^{- 1} (\frac{3}{5}) \begin{array}{l} = \sin^{- 1} (\frac{12}{13} \sqrt{1 - {(\frac{3}{5})}^{2}} - \frac{3}{5} \sqrt{1 - {(\frac{12}{13})}^{2}}) \\ [∵ \sin^{- 1} x - \sin^{- 1} y = \sin^{- 1} (x \sqrt{1 - y^{2}} - y \sqrt{1 - x^{2}}) \end{array} if x^{2} + y^{2} \leq 1 or if xy > 0 and x^{2} + y^{2} > 1 \forall x, y \in [- 1, 1]] \begin{array}{l} = \sin^{- 1} (\frac{12}{13} \times \frac{4}{5} - \frac{3}{5} \times \frac{5}{13}) = \sin^{- 1} (\frac{48 - 15}{65}) \\ = \sin^{- 1} (\frac{33}{65}) = \cos^{- 1} \sqrt{1 - {(\frac{33}{65})}^{2}} \\ = \cos^{- 1} \sqrt{\frac{3136}{4225}} [∵ \sin^{- 1} x = \cos^{- 1} \sqrt{1 - x^{2}}] \\ = \cos^{- 1} (\frac{56}{65}) \\ = \frac{π}{2} - \sin^{- 1} (\frac{56}{65}) [∵ \sin^{- 1} θ + \cot^{- 1} θ = \frac{π}{2}] \end{array}$

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