First slide

Properties of ITF

Question

The value of $\cos (\sin^{- 1} \frac{1}{4} + \sec^{- 1} \frac{4}{3})$ , is

Moderate

A

$\frac{3 \sqrt{15} - \sqrt{7}}{16}$
B

$\frac{3 \sqrt{15} + \sqrt{7}}{16}$
C

$\frac{\sqrt{7} - 3 \sqrt{15}}{16}$
D

$\frac{3 \sqrt{15} - \sqrt{7}}{4}$

Solution

$\cos (\sin^{- 1} \frac{1}{4} + \sec^{- 1} \frac{4}{3})$
$= \cos (\sin^{- 1} \frac{1}{4} + \cos^{- 1} \frac{3}{4})$
$\begin{aligned} = \cos (\sin^{- 1} \frac{1}{4}) \cos (\cos^{- 1} \frac{3}{4}) - \sin (\sin^{- 1} \frac{1}{4}) \sin (\cos^{- 1} \frac{3}{4}) \\ = \cos (\cos^{- 1} \frac{\sqrt{15}}{4}) \cos (\cos^{- 1} \frac{3}{4}) - \sin (\sin^{- 1} \frac{1}{4}) \sin (\sin^{- 1} \frac{\sqrt{7}}{4}) \\ [∵ \sin^{- 1} \frac{1}{4} = \cos^{- 1} \frac{\sqrt{15}}{4} and \cos^{- 1} \frac{3}{4} = \sin^{- 1} \frac{\sqrt{7}}{4}] \\ = \frac{\sqrt{15}}{4} \times \frac{3}{4} - \frac{1}{4} \times \frac{\sqrt{7}}{4} = \frac{3 \sqrt{15} - \sqrt{7}}{16} \end{aligned}$

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