If ^{-1} x = ^{-1} x, then which one of the following is correct?

Correct Answer is : x=12x={1}{{2}}x=21

Correct Answer is : x=12x={1}{{2}}x=21

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UPSC NDA I 2024 Mathematics Paper

Question : 39 of 120

Marks: +1, -0

\cos^{-1} x = \sin^{-1} x

, then which one of the following is correct?

Solution:

Concept:

Use the identity

\sin^{-1} x = \cos^{-1} \sqrt{1 - x^2}

for

x \in [0, 1]

to equate the given inverse trigonometric functions.

Explanation:

Given

\cos^{-1} x = \sin^{-1} x

Since both angles are equal, we can write

\sin^{-1} x = \cos^{-1} \sqrt{1 - x^2}

for

x \geq 0

Thus,

\cos^{-1} x = \cos^{-1} \sqrt{1 - x^2}

Equating the arguments:

x = \sqrt{1 - x^2}

Square both sides:

x^2 = 1 - x^2

2x^2 = 1

, hence

x^2 = \frac{1}{2}

Taking the positive root (since principal values of

\sin^{-1}

and

\cos^{-1}

lie in

[0,\pi/2]

for

x>0

), we get

x = \frac{1}{\sqrt{2}}

Answer:

Option C:

x = \frac{1}{\sqrt{2}}

is correct.

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