NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

The value of $\tan 1^\circ \tan 2^\circ \tan 3^\circ \dots \tan 89^\circ$ is:

Choose the correct answer:

Explanation

Solution

Concept:

To solve this, we use the following trigonometric properties:

$\tan(90^\circ - \theta) = \cot \theta$
$\tan \theta \times \cot \theta = 1$
$\tan 45^\circ = 1$

Calculation:

The given expression is:

E = \tan 1^\circ \tan 2^\circ \dots \tan 44^\circ \tan 45^\circ \tan 46^\circ \dots \tan 88^\circ \tan 89^\circ

We can rewrite the terms from $46^\circ$ to $89^\circ$ using the complementary angle identity:

E = (\tan 1^\circ \tan 2^\circ \dots \tan 44^\circ) \tan 45^\circ [\tan(90^\circ - 44^\circ) \dots \tan(90^\circ - 2^\circ) \tan(90^\circ - 1^\circ)]

Applying the identity $\tan(90^\circ - \theta) = \cot \theta$ :

E = (\tan 1^\circ \tan 2^\circ \dots \tan 44^\circ) \tan 45^\circ (\cot 44^\circ \dots \cot 2^\circ \cot 1^\circ)

Now, rearranging the terms to pair $\tan \theta$ with $\cot \theta$ :

E = (\tan 1^\circ \cot 1^\circ) (\tan 2^\circ \cot 2^\circ) \dots (\tan 44^\circ \cot 44^\circ) \tan 45^\circ

Since $\tan \theta \cot \theta = 1$ and $\tan 45^\circ = 1$ , we get:

E = (1) \times (1) \times \dots \times (1) \times 1

E = 1

Correct Option: 3