Evaluate $\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$

In the given trigonometric expression, cosine and sine functions are involved. Two cosine functions with $100$ degrees and $40$ degrees are multiplied and two sine functions with $100$ degrees and $40$ degrees are also multiplied. The summation of them has to evaluate in this trigonometry problem.

$\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$

Simplify the trigonometric expression

The values of cosine of $100$ degrees, cosine of $40$ degrees, sine of $100$ degrees and sine of $40$ degrees are unknown. So, it is recommendable to simplify the given trigonometric expression for evaluating its value. So, let’s analyze or analyse the possibility of simplifying the given trigonometric expression.

$\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$

1. Two cosine functions get multiplied and two sine functions get multiplied in the given trigonometric expression.
2. Both cosines and sines functions contain the same angles $100$ degrees and $40$ degrees.
3. The products of trigonometric functions are connected by a plus sign for forming a trigonometric expression in this problem.

The three notable factors help us to simplify the given trigonometric expression and these factors made the given trigonometric expression to represent the cosine of angle difference trigonometric formula. Hence, we can simplify the given trigonometric expression by the cos angle difference identity.

$\implies$ $\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$ $\,=\,$ $\cos{(100^\circ-40^\circ)}$

We can now simplify the trigonometric expression further by finding the difference between the angles in the cosine function.

$\,\,\,=\,\,\,\,\,\,$ $\cos{(100^\circ-40^\circ)}$

$\,\,\,=\,\,\,\,\,\,$ $\cos{(60^\circ)}$

Evaluate the cosine of angle

The cosine of sixty degrees is known to us and it is equal to the quotient of $1$ by $2$.

$\,\,\,=\,\,\,\,\,\,$ $\dfrac{1}{2}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$ $\,=\,$ $\dfrac{1}{2}$