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  4. Value of sin 47°+ sin 61°- sin 11°- sin 25° is

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Q. Value of $\sin 47^{\circ}+\sin 61^{\circ}-\sin 11^{\circ}-\sin 25^{\circ}$ is

Trigonometric Functions

Solution:

Given value $=\left(\sin 47^{\circ}+\sin 61^{\circ}\right)-\left(\sin 11^{\circ}+\sin 25^{\circ}\right)$
$=2 \sin 54^{\circ} \cos 7^{\circ}-2 \sin 18^{\circ} \cos 7^{\circ}$
$=2 \cos 7^{\circ}\left(\sin 54^{\circ}-\sin 18^{\circ}\right)=2 \cos 7^{\circ} \,2 \cos 36^{\circ} \sin 18^{\circ}$
$=2 \cos 7^{\circ} \frac{2 \sin 18^{\circ} \cos 18^{\circ}}{\cos 18^{\circ}} \times \cos 36^{\circ}$
$=\cos 7^{\circ} \frac{2 \sin 36^{\circ} \cos 36^{\circ}}{\cos 18^{\circ}}=\cos 7^{\circ} \frac{\sin 72^{\circ}}{\cos 18^{\circ}}=\cos 7^{\circ}$
$\left[\because \sin 72^{\circ}=\cos 18^{\circ}\right]$