Evaluate sin 78° cos 18° − cos 78° sin 18°

Question

Evaluate:

\[ \sin 78^\circ \cos 18^\circ-\cos 78^\circ \sin 18^\circ \]

Solution

Using the identity:

\[ \sin A \cos B-\cos A \sin B=\sin(A-B) \]

Here,

\[ A=78^\circ,\qquad B=18^\circ \]

Therefore,

\[ \sin 78^\circ \cos 18^\circ-\cos 78^\circ \sin 18^\circ \]

\[ =\sin(78^\circ-18^\circ) \]

\[ =\sin 60^\circ \]

We know that:

\[ \sin 60^\circ=\frac{\sqrt{3}}{2} \]

Therefore,

\[ \boxed{\sin 78^\circ \cos 18^\circ-\cos 78^\circ \sin 18^\circ=\frac{\sqrt{3}}{2}} \]