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UPSC NDA I 2021 Mathematics Solved Paper

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Question : 34 of 120

Marks: +1, -0

What is the value of the following?

\tan 31^{\circ} \tan 33^{\circ} \tan 35^{\circ} \dots \tan 57^{\circ} \tan 59^{\circ}

$-1$
0
1
2

Solution:

Concept:

Use the identity

\tan \theta = \cot (90^{\circ} - \theta)

and

\tan \theta \cdot \cot \theta = 1

to pair terms in the product.

Explanation:

The product is

\tan 31^{\circ} \tan 33^{\circ} \tan 35^{\circ} \dots \tan 57^{\circ} \tan 59^{\circ}

.

The angles increase by

2^{\circ}

from

31^{\circ}

to

59^{\circ}

.

Rewrite each term after the midpoint using

\tan \theta = \cot (90^{\circ} - \theta)

.

For example,

\tan 59^{\circ} = \cot 31^{\circ}

,

\tan 57^{\circ} = \cot 33^{\circ}

, and so on.

The product becomes:

(\tan 31^{\circ} \cdot \cot 31^{\circ}) \times (\tan 33^{\circ} \cdot \cot 33^{\circ}) \times \dots \times \tan 45^{\circ}

.

Each pair

\tan \theta \cdot \cot \theta = 1

.

The middle term is

\tan 45^{\circ} = 1

.

Thus, the entire product equals

1

.

Answer:

1

(Option C)

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