If y= ²≤ft( ^{-1} {1+x/1-x}), then d y/d x has the value

Correct Answer is : −12-1/22−1

Correct Answer is : −12-1/22−1

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MHTCET 10 May 2024 Shift 2 Paper

Question : 125 of 150

Marks: +1, -0

y=\sin ^{2}\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)

\frac{d y}{d x}

has the value

Solution:

Concept:

Simplify the nested trigonometric expression using

\cot^2 \theta

and then differentiate the resulting simple function.

Explanation:

Let

\theta = \cot^{-1}\sqrt{\frac{1+x}{1-x}}

Then

\cot^2 \theta = \frac{1+x}{1-x}

Add

1

to both sides:

1 + \cot^2 \theta = 1 + \frac{1+x}{1-x} = \frac{2}{1-x}

Using

1 + \cot^2 \theta = \csc^2 \theta

, we get:

\csc^2 \theta = \frac{2}{1-x}

Therefore,

\sin^2 \theta = \frac{1-x}{2}

Since

y = \sin^2 \theta

, we substitute directly:

y = \frac{1-x}{2}

Differentiate with respect to

x

\frac{dy}{dx} = -\frac{1}{2}

Answer:

\frac{dy}{dx} = -\frac{1}{2}

, so the correct option is A.

\frac{-1}{2}

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