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UPSC NDA I 2022 Mathematics Solved Paper

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Question : 26 of 120

Marks: +1, -0

\text{What is value of } \cot^{2} 15^{\circ} + \tan^{2} 15^{\circ} ?

12
14
$8\sqrt{3}$
4

Solution:

Concept:

Use the exact known values:

\tan 15^\circ = 2 - \sqrt{3}

and

\cot 15^\circ = 2 + \sqrt{3}

.

Then compute the sum of their squares.

Explanation:

Given expression:

\cot^2 15^\circ + \tan^2 15^\circ

.

Substitute the standard values:

\cot 15^\circ = 2 + \sqrt{3}

and

\tan 15^\circ = 2 - \sqrt{3}

.

Square each term:

(2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}

.

And

(2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}

.

Add them:

(7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) = 14

.

Answer:

14

(Option B).

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