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UPSC NDA 2 2022 Mathematics Solved Paper

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Question : 53 of 120

Marks: +1, -0

What is the value of tan2 165° + cot2 165° ?

7
14
4 $\sqrt{3}$
8 $\sqrt{3}$

Solution:

Concept:

\tan(180^\circ - \theta) = -\tan\theta

and

\cot(180^\circ - \theta) = -\cot\theta

.

Standard values:

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

,

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

.

Explanation:

Given expression:

\tan^2 165^\circ + \cot^2 165^\circ

.

Since

165^\circ = 180^\circ - 15^\circ

, we have:

\tan 165^\circ = \tan(180^\circ - 15^\circ) = -\tan 15^\circ

, so

\tan^2 165^\circ = \tan^2 15^\circ

.

Similarly,

\cot 165^\circ = \cot(180^\circ - 15^\circ) = -\cot 15^\circ

, so

\cot^2 165^\circ = \cot^2 15^\circ

.

Thus the expression becomes

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

.

Substitute the known values:

(2 - \sqrt{3})^2 + (2 + \sqrt{3})^2

.

Compute each square:

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

.

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

.

Add:

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

.

Answer:

The value is

14

. Hence, option B is correct.

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