Question
Evaluate the following: $\cot 20^\circ \cot 40^\circ \cot 45^\circ \cot 50^\circ \cot 70^\circ $
✓
Answer
$ \cot 20^{\circ} \cot 40^{\circ} \cot 45^{\circ} \cot 50^{\circ} \cot 70^{\circ}$
$ =\cot \left(90^{\circ}-70^{\circ}\right) \times \cot \left(90^{\circ}-50^{\circ}\right) \times 1 \times \cot 50^{\circ} \times \cot 70^{\circ}$
$ =\tan 70^{\circ} \times \tan 50^{\circ} \times \cot 50^{\circ} \times \cot 70^{\circ}$
$ =\tan 70^{\circ} \times \cot 70^{\circ} \times \tan 50^{\circ} \times \cot 50^{\circ}$
$ =\tan 70^{\circ} \times \frac{1}{\tan 70^{\circ}} \times \tan 50^{\circ} \times \frac{1}{\tan 50^{\circ}}$
$ =1 .$
$ =\cot \left(90^{\circ}-70^{\circ}\right) \times \cot \left(90^{\circ}-50^{\circ}\right) \times 1 \times \cot 50^{\circ} \times \cot 70^{\circ}$
$ =\tan 70^{\circ} \times \tan 50^{\circ} \times \cot 50^{\circ} \times \cot 70^{\circ}$
$ =\tan 70^{\circ} \times \cot 70^{\circ} \times \tan 50^{\circ} \times \cot 50^{\circ}$
$ =\tan 70^{\circ} \times \frac{1}{\tan 70^{\circ}} \times \tan 50^{\circ} \times \frac{1}{\tan 50^{\circ}}$
$ =1 .$
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