The equation ^{-1}x - ^{-1}x = π/6 has

Correct Answer is : unique solution

Correct Answer is : unique solution

Test Index

UPSC NDA I 2021 Mathematics Solved Paper

Question : 36 of 120

Marks: +1, -0

The equation

\sin^{-1}x - \cos^{-1}x = \frac{\pi}{6}

has

Solution:

Concept:

Use the identity

\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}

to simplify the equation.

Explanation:

Given

\sin^{-1}x - \cos^{-1}x = \frac{\pi}{6}

Rewrite as

(\sin^{-1}x + \cos^{-1}x) - 2\cos^{-1}x = \frac{\pi}{6}

Substitute the identity:

\frac{\pi}{2} - 2\cos^{-1}x = \frac{\pi}{6}

Solve for

\cos^{-1}x

2\cos^{-1}x = \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}

Thus

\cos^{-1}x = \frac{\pi}{6}

This gives

x = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}

The value is within the domain

[-1,1]

, so there is exactly one solution.

Answer:

Option B: unique solution.

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