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UPSC NDA I 2019 Mathematics Solved Paper

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Question : 43 of 120

Marks: +1, -0

What is the value of tan 75° + cot 75°?

2
4
2√3
4√3

Solution:

Concept:

Use the tangent addition formula and the reciprocal identity for cotangent.

Explanation:

We need

\tan 75^\circ + \cot 75^\circ

.

First, find

\tan 75^\circ

using

\tan(45^\circ + 30^\circ)

.

\tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}

.

\tan 45^\circ = 1

and

\tan 30^\circ = \frac{1}{\sqrt{3}}

.

So

\tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} = \frac{\sqrt{3}+1}{\sqrt{3}-1}

.

Since

\cot x = \frac{1}{\tan x}

, we have

\cot 75^\circ = \frac{\sqrt{3}-1}{\sqrt{3}+1}

.

Now add:

\frac{\sqrt{3}+1}{\sqrt{3}-1} + \frac{\sqrt{3}-1}{\sqrt{3}+1}

.

Combine into a single fraction:

\frac{(\sqrt{3}+1)^2 + (\sqrt{3}-1)^2}{(\sqrt{3}-1)(\sqrt{3}+1)}

.

Numerator:

(3 + 2\sqrt{3} + 1) + (3 - 2\sqrt{3} + 1) = 8

.

Denominator:

(\sqrt{3})^2 - 1^2 = 3 - 1 = 2

.

Thus

\frac{8}{2} = 4

.

Answer:

4

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