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  4. The value of cot 70°+4 cos 70° is

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Q. The value of $\cot 70^{\circ}+4 \cos 70^{\circ}$ is

Trigonometric Functions

Solution:

$\cot 70^{\circ}+4 \cos 70^{\circ} =\frac{\cos 70^{\circ}+4 \sin 70^{\circ} \cos 70^{\circ}}{\sin 70^{\circ}}$
$=\frac{\cos 70^{\circ}+2 \sin 140^{\circ}}{\sin 70^{\circ}}$
$=\frac{\cos 70^{\circ}+2 \sin \left(180^{\circ}-40^{\circ}\right)}{\sin 70^{\circ}}$
$=\frac{\sin 20^{\circ}+\sin 40^{\circ}+\sin 40^{\circ}}{\sin 70^{\circ}}$
$=\frac{2 \sin 30^{\circ} \cos 10^{\circ}+\sin 40^{\circ}}{\sin 70^{\circ}}$
$=\frac{\sin 80^{\circ}+\sin 40^{\circ}}{\sin 70^{\circ}}$
$=\frac{2 \sin 60^{\circ} \cos 20^{\circ}}{\sin 70^{\circ}}$
$=\sqrt{3}$