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UPSC CDS 1 2020 Math Paper

Question : 52 of 100

Marks: +1, -0

(25\csc^{2}x + \sec^{2}x)

Solution:

Concept:

Use trigonometric identities to express the expression in terms of

\tan x

and

\cot x

, then apply AM‑GM inequality to find the minimum.

Explanation:

Rewrite

25\csc^{2}x + \sec^{2}x

using the identities

\csc^{2}x = 1 + \cot^{2}x

and

\sec^{2}x = 1 + \tan^{2}x

So the expression becomes

25(1 + \cot^{2}x) + (1 + \tan^{2}x) = 26 + \left(25\cot^{2}x + \tan^{2}x\right)

Both

25\cot^{2}x

and

\tan^{2}x

are positive.

By the AM‑GM inequality:

\frac{25\cot^{2}x + \tan^{2}x}{2} \ge \sqrt{25\cot^{2}x \cdot \tan^{2}x} = \sqrt{25} = 5

Thus

25\cot^{2}x + \tan^{2}x \ge 10

Therefore the minimum value of the whole expression is

26 + 10 = 36

Answer:

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