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UPSC NDA I 2023 Mathematics Solved Paper

Question : 94 of 120

Marks: +1, -0

What is the derivative of cosec(x°) ?

-cosec(x°) cot(x°)
$-\frac{\pi }{180}cosec(x^{∘})cot(x^{∘})$
$\frac{\pi }{180}cosec(x^{∘})cot(x^{∘})$
$-\frac{\pi }{180}cosec(x)cot(x)$

Solution:

Concept:

To differentiate a trigonometric function when the angle is in degrees, first convert the angle to radians because calculus formulas assume radian measure.

Explanation:

Let

y = \csc(x^\circ)

Since

x^\circ = \frac{x\pi}{180}

radians, we have

y = \csc\left(\frac{x\pi}{180}\right)

Differentiate using the chain rule:

\frac{d}{dx}\csc u = -\csc u \cot u \cdot \frac{du}{dx}

Here

u = \frac{x\pi}{180}

, so

\frac{du}{dx} = \frac{\pi}{180}

Thus

\frac{dy}{dx} = -\csc\left(\frac{x\pi}{180}\right) \cot\left(\frac{x\pi}{180}\right) \cdot \frac{\pi}{180}

Since

\csc\left(\frac{x\pi}{180}\right) = \csc(x^\circ)

and

\cot\left(\frac{x\pi}{180}\right) = \cot(x^\circ)

, the derivative is

-\frac{\pi}{180}\csc(x^\circ)\cot(x^\circ)

Answer:

-\frac{\pi}{180}\csc(x^\circ)\cot(x^\circ)

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