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Question: How do you write the expression \[\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \c...

How do you write the expression cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } as sine, cosine, or tangent of an angle?

Explanation

Solution

This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. To solve this problem we need to know the basic trigonometric identities. We need to know how to compare the given expression with trigonometric identities to make an easy calculation. The final answer would be a simplified form of the given expression.

Complete step by step solution:
The given expression in the 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)
We know that,
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B
The above equation can also be written as,
cos⁡Acos⁡B−sin⁡Asin⁡B=cos⁡(A+B)→(2)\cos A\cos B - \sin A\sin B = \cos \left( {A + B} \right) \to \left( 2 \right)
Let’s compare the equation (1)\left( 1 \right)and(2)\left( 2 \right), we get
(1)→cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘\left( 1 \right) \to \cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ }
(2)→cos⁡Acos⁡B−sin⁡Asin⁡B=cos⁡(A+B)\left( 2 \right) \to \cos A\cos B - \sin A\sin B = \cos \left( {A + B} \right)
By comparing these two equations we get,
The value of AA is 25∘{25^ \circ } and the value of BB is 15∘{15^ \circ }
So, the equation (2)\left( 2 \right) can also be written as,
(2)→cos⁡Acos⁡B−sin⁡Asin⁡B=cos⁡(A+B)\left( 2 \right) \to \cos A\cos B - \sin A\sin B = \cos \left( {A + B} \right)
cos⁡25∘cos⁡15∘−sin⁡25∘sin⁡15∘=cos⁡(25∘+15∘)\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = \cos \left( {{{25}^ \circ } + {{15}^ \circ }} \right)
By using addition operation to solve the above equation we get,

\right)$$ **So, the final answer is, $$\cos {25^ \circ }\cos {15^ \circ } - \sin {25^ \circ }\sin {15^ \circ } = \cos \left( {{{35}^ \circ }} \right)$$** We won’t find the value $$\cos \left( {{{35}^ \circ }} \right)$$ from the above equation, because in this question they ask the final answer in sine, cosine, or tangent of an angle. Here angle is the cosine of $${35^ \circ }$$. **Note:** This question describes the operation of arithmetic functions like addition/ subtraction/ multiplication/ division. Remember the basic trigonometric identities to solve these types of questions. Note that the final answer would contain angle instead of numbers for these types of questions. So we don’t need to calculate the cosine angle in the final answer. Because in this question they ask the final answer in the form of sine, cosine, or tangent of an angle. Also, note that $$\cos \left( { - \theta } \right)$$ it also can be written as $$\cos \theta $$.