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Question: Evaluate the following:- \[\cos {{80}^{\circ }}\cdot \cos {{20}^{\circ }}+\sin {{80}^{\circ }}\cdo...

Evaluate the following:-
cos⁡80∘⋅cos⁡20∘+sin⁡80∘⋅sin⁡20∘\cos {{80}^{\circ }}\cdot \cos {{20}^{\circ }}+\sin {{80}^{\circ }}\cdot \sin {{20}^{\circ }}

Explanation

Solution

Hint:In such questions, we try to find the angles that are involved in the question and then we try to convert it into a known angle by subtracting or adding the involved angles.

Complete step-by-step answer:
In this question, we would be using the following formula that is required to evaluate the sum of angles within a cos function.
cos⁡(a−b)=cos⁡a⋅cos⁡b+sin⁡a⋅sin⁡b\cos (a-b)=\cos a\cdot \cos b+\sin a\cdot \sin b
(Where a and b are two separate angles)

As mentioned in the question, we have to find the value of the expression that is mentioned.
Now, as mentioned in the hint, we are going to find the value of the angles that are involved in the expression.
We can observe that in the given expression, we get the angles that are involved as follows
80∘ and 20∘{{80}^{\circ }}\ and\ {{20}^{\circ }}
Now, we can see that the difference of these two involved angles is 60∘{{60}^{\circ }} , which is a known angle. So, we will use the formula that is given in the hint to find the value of the expression that is as follows

& =\cos {{80}^{\circ }}\cdot \cos {{20}^{\circ }}+\sin {{80}^{\circ }}\cdot \sin {{20}^{\circ }} \\\ & =\cos ({{80}^{\circ }}-{{20}^{\circ }}) \\\ & =\cos {{60}^{\circ }} \\\ & =\dfrac{1}{2} \\\ \end{aligned}$$ (Using the formula that is given in the hint) Hence, the value of the given expression is $$\dfrac{1}{2}$$ . Note:The students can make an error if they don’t know how to solve the expression and the formula that is required to solve this question that is also mentioned in the hint as follows $$\cos (a-b)=\cos a\cdot \cos b+\sin a\cdot \sin b$$ (Where a and b are two separate angles)