$\tan 1^{\circ} \times \tan 2^{\circ} \times \tan 3^{\circ} \times \cdots \cdots \cdots+\cdots \times \tan…

$\tan 1^{\circ} \times \tan 2^{\circ} \times \tan 3^{\circ} \times \cdots \cdots \cdots+\cdots \times \tan 89^{\circ}=$
  1. $\sqrt{3}$
  2. 1
  3. $\sqrt{2}$
  4. 2

Solution

$\tan 1^{\circ} \times \tan 2^{\circ} \times \tan 3^{\circ} \times \ldots \times \tan 89^{\circ}$ $=\left[\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 44^{\circ}\right]\left(\tan 45^{\circ}\right) \times\left[\tan \left(90^{\circ}-44^{\circ}\right) \cdot \tan \left(90^{\circ}-43^{\circ}\right) \ldots \tan \left(90^{\circ}-1\right)\right]$ $=\left(\tan 1^{\circ} \tan 2^{\circ} \ldots \tan 44^{\circ}\right)\left(\cot 44^{\circ} \cot 43^{\circ} \ldots \cot 1^{\circ}\right)$ $=1$

Asked in: MHT CET 2020 (12 Oct Shift 1)

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