Evaluate cos 47° cos 13° − sin 47° sin 13°

Question

Evaluate:

\[ \cos 47^\circ \cos 13^\circ-\sin 47^\circ \sin 13^\circ \]

Solution

Using the identity:

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

Here,

\[ A=47^\circ,\qquad B=13^\circ \]

Therefore,

\[ \cos 47^\circ \cos 13^\circ-\sin 47^\circ \sin 13^\circ \]

\[ =\cos(47^\circ+13^\circ) \]

\[ =\cos 60^\circ \]

We know that:

\[ \cos 60^\circ=\frac{1}{2} \]

Therefore,

\[ \boxed{\cos 47^\circ \cos 13^\circ-\sin 47^\circ \sin 13^\circ=\frac{1}{2}} \]