First slide

Trigonometric transformations

Question

$\cot 16^{\circ} \cot 44^{\circ} + \cot 44^{\circ} \cot 76^{\circ} - \cot 76^{\circ} \cot 16^{\circ} =$

Moderate

A

1
B

2
C

3
D

4

Solution

$\begin{matrix} \frac{\cos 16^{\circ} \cos 44^{\circ}}{\sin 16^{\circ} \sin 44^{\circ}} - 1 + \frac{\cos 44^{\circ} \cos 76^{\circ}}{\sin 44^{\circ} \sin 76^{\circ}} - 1 - \frac{\cos 76^{\circ} \cos 16^{\circ}}{\sin 76^{\circ} \sin 16^{\circ}} - 1 + 3 \\ = \frac{\cos 60^{\circ}}{\sin 16^{\circ} \sin 44^{\circ}} + \frac{\cos 120^{\circ}}{\sin 44^{\circ} \sin 76^{\circ}} - \frac{\cos 60^{\circ}}{\sin 76^{\circ} \sin 16^{\circ}} + 3 \\ = \frac{1}{2} (\frac{\sin 76^{\circ} - \sin 16^{\circ}}{\sin 16^{\circ} \sin 44^{\circ} \sin 76^{\circ}}) - \frac{1}{2 \sin 76^{\circ} \sin 16^{\circ}} + 3 \\ = \frac{1}{2} (\frac{2 \sin 30^{\circ} \cos 46^{\circ}}{\sin 16^{\circ} \sin 44^{\circ} \sin 76^{\circ}}) - \frac{1}{2 \sin 76^{\circ} \sin 16^{\circ}} + 3 \\ = \frac{1}{2} (\frac{2 \sin 30^{\circ} \sin 44^{\circ}}{\sin 16^{\circ} \sin 44^{\circ} \sin 76^{\circ}}) - \frac{1}{2 \sin 76^{\circ} \sin 16^{\circ}} + 3 \\ = 3 \end{matrix}$

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