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  4. The value of ( tan 70°- tan 20°/ tan 50°)=

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Q. The value of $\frac{\tan\, 70^{\circ}-\tan \,20^{\circ}}{\tan \,50^{\circ}}=$

Trigonometric Functions

Solution:

$\frac{\tan \,70^{\circ}-\tan \,20^{\circ}}{\tan \,50^{\circ}}=\frac{\frac{\sin \,70^{\circ}}{\cos\, 70^{\circ}}-\frac{\sin \,20^{\circ}}{\cos \,20^{\circ}}}{\frac{\sin \,50^{\circ}}{\cos \,50^{\circ}}}$
$\frac{\frac{sin \,70^{\circ} cos \,20^{\circ}-cos\, 70^{\circ} sin\, 20^{\circ}}{cos \,70^{\circ}cos\, 20^{\circ}}}{\frac{sin \,50^{\circ}}{cos\, 50^{\circ}}}$
$=\frac{2}{2} \times \frac{\sin \left(70^{\circ}-20^{\circ}\right) \cos 50^{\circ}}{\cos 70^{\circ} \cos 20^{\circ} \sin 50^{\circ}}=\frac{2 \sin 50^{\circ} \cos 50^{\circ}}{2 \cos 70^{\circ} \cos 20^{\circ} \sin 50^{\circ}}$
$=\frac{2 \cos 50^{\circ}}{\cos 90^{\circ}+\cos 50^{\circ}}=\frac{2 \cos 50^{\circ}}{0+\cos 50^{\circ}}=2$