Find the exact value of $\cos 80^{\circ} \cos 20^{\circ}+\sin 80^{\circ} \sin 20^{\circ}$

Step 1 ：Given the expression \(\cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ\)

Step 2 ：This is a standard trigonometric identity. The formula for the cosine of the difference of two angles is given by: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\)

Step 3 ：In this case, a = 80 and b = 20. So, the expression \(\cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ\) is equivalent to \(\cos(80^\circ - 20^\circ)\)

Step 4 ：Calculating the value of \(\cos(80^\circ - 20^\circ)\) gives approximately 0.5, which is the expected value for \(\cos 60^\circ\)

Step 5 ：This confirms that the original expression is indeed equivalent to \(\cos(80^\circ - 20^\circ)\)

Step 6 ：Final Answer: The exact value of \(\cos 80^\circ \cos 20^\circ+\sin 80^\circ \sin 20^\circ\) is \(\boxed{0.5}\)