Evaluate cos 80° cos 20° + sin 80° sin 20°

Question

Evaluate:

\[ \cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ \]

Solution

Using the identity:

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

Here,

\[ A=80^\circ,\qquad B=20^\circ \]

Therefore,

\[ \cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ \]

\[ =\cos(80^\circ-20^\circ) \]

\[ =\cos 60^\circ \]

We know that:

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

Therefore,

\[ \boxed{\cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ=\frac{1}{2}} \]