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The value of \( \left(\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}\right) \) is
(A) 0
(B) 1
(C) 2
(D) \( \frac{1}{2} \)
Given:
\( \left(\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}\right) \)
To do:
We have to find the value of \( \left(\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}\right) \).
Solution:
We know that,
$tan\ (90^{\circ}- \theta) = cot\ \theta$
$tan\ \theta \times \cot\ \theta=1$
Therefore,
$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}.....\tan 45^{\circ}...... \tan 87^{\circ} \tan 88^{\circ} \tan 89^{\circ}=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}......\tan 45^{\circ}.........\tan (90^{\circ}-3^{\circ}) \tan (90^{\circ}-2^{\circ}) \tan (90^{\circ}-1^{\circ})$
$=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}........\tan 45^{\circ}......\cot 3^{\circ} \cot 2^{\circ} \cot 1^{\circ}$
$=(\tan 1^{\circ} \cot 1^{\circ})(\tan 2^{\circ} \cot 2^{\circ})..................(\tan 44^{\circ} \cot 44^{\circ})(1)$
$=1$