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Question: How do you find the exact functional value \[\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\...

How do you find the exact functional value cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ }using the cosine sum (or) difference identity?

Explanation

Solution

The given question describes the operation of using addition/ subtraction/ multiplication/ division. Also, in this question, we would use the trigonometric formula related to the given question. To solve the given question we would compare the given equation with the cosine sum (or) difference identity. After comparing the two-equation we would find the value of variables in the equation. Also, remind the basic trigonometric table value and related formula to find the value of the given equation.

Complete answer:
The given question is shown below,
cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘=?→(1)\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = ? \to \left( 1 \right)
To simplify the given question we have to compare the above equation with the cosine sum (or) difference identity which is given below,
cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos (a + b) = \cos a\cos b - \sin a\sin b
It also can be written as,
cos⁡acos⁡b−sin⁡asin⁡b=cos⁡(a+b)→(2)\cos a\cos b - \sin a\sin b = \cos (a + b) \to \left( 2 \right)
Let’s compare the equation (1) and (2), we get

(1)→cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘=? (2)→cos⁡acos⁡b−sin⁡asin⁡b=cos⁡(a+b)  \left( 1 \right) \to \cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = ? \\\ \left( 2 \right) \to \cos a\cos b - \sin a\sin b = \cos (a + b) \\\

By comparing the above two equations we get, the value of aa is 2525 and the value of bb is1515.
By substituting the value of aa and bb in the equation(2)(2), we get
cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘=cos⁡(25∘+15∘)\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = \cos ({25^ \circ } + {15^ \circ })
It also can be written as,
cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘=cos⁡(40∘)\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = \cos ({40^ \circ })
So, we get
cos⁡(40∘)=0.766\cos ({40^ \circ }) = 0.766(Using calculator in degree mode)
So, the final answer is cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘=0.766\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = 0.766

Note: In this type of question we would use the operation of addition/ subtraction/ multiplication/ division. After that, we would try to compare the given question with the cosine sum (or) difference identity. When finding cos⁡\cos value in the degree we should select the degree mode in the calculator. Also, note that cos⁡(−θ)\cos \left( { - \theta } \right) is cos⁡(θ)\cos \left( \theta \right) and sin⁡(−θ)\sin ( - \theta ) is sin⁡(θ)\sin (\theta ). If the θ\theta value is in decimal we should use the calculator in radian mode.