Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# The value of tan 1 ∘ tan 2 ∘ tan 3 ∘ . . . tan 89 ∘ is - Mathematics

MCQ

The value of $\tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ$ is

#### Options

• 0

• 1

• $\frac{1}{2}$

• not defined

#### Solution

We know that,
$\tan\left( 90^\circ - \theta \right) = \cot\theta$
So,

$\tan89^\circ = \tan\left( 90^\circ - 1^\circ \right) = \cot1^\circ$
$\tan88^\circ = \tan\left( 90^\circ - 2^\circ \right) = \cot2^\circ$
$\tan87^\circ = \tan\left( 90^\circ - 3^\circ \right) = \cot3^\circ$
. . . .
. . . .
$\tan46^\circ = \tan\left( 90^\circ - 44^\circ \right) = \cot44^\circ$

$\therefore \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ$

$= \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan44^\circ \tan45^\circ \tan46^\circ . . . \tan87^\circ \tan88^\circ \tan89^\circ$

$= \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan44^\circ \tan45^\circ \cot44^\circ. . . \cot3^\circ \cot2^\circ \cot1^\circ$

$= \left( \tan1^\circ\cot1^\circ \right)\left( \tan2^\circ\cot2^\circ \right) \left( \tan3^\circ\cot3^\circ \right) . . . \left( \tan44^\circ\cot44^\circ \right)\tan45^\circ$

$= 1 \left( \tan45^\circ = 1\text{ and }\tan\theta\cot\theta = 1 \right)$

Hence, the correct answer is option 1.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 5 Trigonometric Functions
Q 27 | Page 43

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