Evaluate the following: 5 5^ 15^ 25^ 35^ 45^ 7 45^ 55^ 65^ 75^ 85… - Vidyadip

Question

Evaluate the following:$\frac{5 \cot 5^{\circ} \cot 15^{\circ} \cot 25^{\circ} \cot 35^{\circ} \cot 45^{\circ}}{7 \tan 45^{\circ} \tan 55^{\circ} \tan 65^{\circ} \tan 75^{\circ} \tan 85^{\circ}}+\frac{2 \operatorname{cosec} 12^{\circ} \operatorname{cosec} 24^{\circ} \cos 78^{\circ} \cos 66^{\circ}}{7 \sin 14^{\circ} \sin 23^{\circ} \sec 76^{\circ} \sec 67^{\circ}}$

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Answer

$ \frac{5 \cot 5^{\circ} \cot 15^{\circ} \cot 25^{\circ} \cot 35^{\circ} \cot 45^{\circ}}{7 \tan 45^{\circ} \tan 55^{\circ} \tan 65^{\circ} \tan 75^{\circ} \tan 85^{\circ}}+\frac{2 \operatorname{cosec} 12^{\circ} \operatorname{cosec} 24^{\circ} \cos 78^{\circ} \cos 66^{\circ}}{7 \sin 14^{\circ} \sin 23^{\circ} \sec 76^{\circ} \sec 67^{\circ}}$
$\frac{5 \cot \left(90^{\circ}-85^{\circ}\right) \cot \left(90^{\circ}-75^{\circ}\right) \cot \left(90^{\circ}-65^{\circ}\right) \cot \left(90^{\circ}-55^{\circ}\right) \times 1}{7 \times 1 \times \tan 55^{\circ} \tan 65^{\circ} \tan 75^{\circ} \tan 85^{\circ}}+\frac{2 \operatorname{cosec}\left(90^{\circ}-78^{\circ}\right) \operatorname{cosec}\left(90^{\circ}-66^{\circ}\right) \cos 78^{\circ} \cos 66^{\circ}}{7 \sin \left(90^{\circ}-76^{\circ}\right) \sin \left(90^{\circ}-67^{\circ}\right) \sec 76^{\circ} \sec 67^{\circ}}$
$ =\frac{5 \tan 85^{\circ} \tan 75^{\circ} \tan 65^{\circ} \tan 55^{\circ}}{7 \times \tan 55^{\circ} \tan 65^{\circ} \tan 75^{\circ} \tan 85^{\circ}}+\frac{2 \sec 78^{\circ} \sec 66^{\circ} \times \frac{1}{\sec 78^{\circ}} \times \frac{1}{\sec 66^{\circ}}}{7 \cos 76^{\circ} \cos 67^{\circ} \times \frac{1}{\cos 76^{\circ}} \times \frac{1}{\cos 67^{\circ}}}$
$ =\frac{5}{7}+\frac{2}{7}$
$ =\frac{7}{7}$
$ =1 \text {. } $

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