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MHTCET 22 Apr 2025 Shift 1 Paper

Question : 135 of 150

Marks: +1, -0

\cot ^{-1}\left(2 \cos \left(2 \operatorname{cosec}^{-1}(\sqrt{2})\right)\right)=\ldots

Solution:

Concept:

Evaluate the nested inverse trigonometric functions step by step using principal values.

Explanation:

Start with the innermost term,

\operatorname{cosec}^{-1}(\sqrt{2})

Since

\csc\left(\frac{\pi}{4}\right) = \sqrt{2}

, the principal value is

\frac{\pi}{4}

Multiply by 2:

2 \times \frac{\pi}{4} = \frac{\pi}{2}

Now evaluate the cosine:

\cos\left(\frac{\pi}{2}\right) = 0

Multiply by the coefficient 2:

2 \times 0 = 0

So the original expression simplifies to

\cot^{-1}(0)

The principal value of

\cot^{-1}(0)

\frac{\pi}{2}

Answer:

\frac{\pi}{2}

, which corresponds to option A.

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