cot⁡16∘cot⁡44∘+cot⁡44∘cot⁡76∘−cot⁡76∘cot⁡16∘=

cos⁡16∘cos⁡44∘sin⁡16∘sin⁡44∘−1+cos⁡44∘cos⁡76∘sin⁡44∘sin⁡76∘−1−cos⁡76∘cos⁡16∘sin⁡76∘sin⁡16∘−1+3 =cos⁡60∘sin⁡16∘sin⁡44∘+cos⁡120∘sin⁡44∘sin⁡76∘−cos⁡60∘sin⁡76∘sin⁡16∘+3 =12sin⁡76∘−sin⁡16∘sin⁡16∘sin⁡44∘sin⁡76∘−12sin⁡76∘sin⁡16∘+3 =122sin⁡30∘cos⁡46∘sin⁡16∘sin⁡44∘sin⁡76∘−12sin⁡76∘sin⁡16∘+3 =122sin⁡30∘sin⁡44∘sin⁡16∘sin⁡44∘sin⁡76∘−12sin⁡76∘sin⁡16∘+3=3

cot⁡16∘cot⁡44∘+cot⁡44∘cot⁡76∘−cot⁡76∘cot⁡16∘=

cos⁡16∘cos⁡44∘sin⁡16∘sin⁡44∘−1+cos⁡44∘cos⁡76∘sin⁡44∘sin⁡76∘−1−cos⁡76∘cos⁡16∘sin⁡76∘sin⁡16∘−1+3 =cos⁡60∘sin⁡16∘sin⁡44∘+cos⁡120∘sin⁡44∘sin⁡76∘−cos⁡60∘sin⁡76∘sin⁡16∘+3 =12sin⁡76∘−sin⁡16∘sin⁡16∘sin⁡44∘sin⁡76∘−12sin⁡76∘sin⁡16∘+3 =122sin⁡30∘cos⁡46∘sin⁡16∘sin⁡44∘sin⁡76∘−12sin⁡76∘sin⁡16∘+3 =122sin⁡30∘sin⁡44∘sin⁡16∘sin⁡44∘sin⁡76∘−12sin⁡76∘sin⁡16∘+3=3

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

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answer is C.

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Detailed Solution

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