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Question: How do you use sum to product formulas to write the sum or difference \(\cos \left( x+\dfrac{\pi }{6...

How do you use sum to product formulas to write the sum or difference cos⁡(x+π6)−cos⁡(x−π6)\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right) as a product?

Explanation

Solution

We first describe the concept and formulas of compound angles for the trigonometric ratio cos. We use the formulas of cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B and cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B. We put the values of A=x;B=π6A=x;B=\dfrac{\pi }{6} to find the solution for cos⁡(x+π6)−cos⁡(x−π6)\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right).

Complete step-by-step answer:
To simplify or to express in product form the given expression cos⁡(x+π6)−cos⁡(x−π6)\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right), we are going to use the laws of compound angles.
The formula for compound angles gives cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B and cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B.
Now we subtract these two formulas to get
cos⁡(A+B)−cos⁡(A−B) =cos⁡Acos⁡B−sin⁡Asin⁡B−cos⁡Acos⁡B−sin⁡Asin⁡B =−2sin⁡Asin⁡B \begin{aligned} & \cos \left( A+B \right)-\cos \left( A-B \right) \\\ & =\cos A\cos B-\sin A\sin B-\cos A\cos B-\sin A\sin B \\\ & =-2\sin A\sin B \\\ \end{aligned}
For our given expression, we use the formula cos⁡(A+B)−cos⁡(A−B)=−2sin⁡Asin⁡B\cos \left( A+B \right)-\cos \left( A-B \right)=-2\sin A\sin B.
We assume the variables as A=x;B=π6A=x;B=\dfrac{\pi }{6}.
Putting the values, we get cos⁡(x+π6)−cos⁡(x−π6)=−2sin⁡xsin⁡(π6)\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right)=-2\sin x\sin \left( \dfrac{\pi }{6} \right).
We know that the value for sin⁡(π6)=12\sin \left( \dfrac{\pi }{6} \right)=\dfrac{1}{2}.
Therefore, we have −2sin⁡xsin⁡(π6)=−2sin⁡x(12)=−sin⁡x-2\sin x\sin \left( \dfrac{\pi }{6} \right)=-2\sin x\left( \dfrac{1}{2} \right)=-\sin x.
Therefore, changing from sum to product we get the simplified form of cos⁡(x+π6)−cos⁡(x−π6)\cos \left( x+\dfrac{\pi }{6} \right)-\cos \left( x-\dfrac{\pi }{6} \right) as −sin⁡x-\sin x.

Note: A compound angle is an algebraic sum of two or more angles. We use trigonometric identities to connote compound angles through trigonometric functions. Similarly, we can change the form from product to sum using the formula of
2sin⁡Asin⁡B=cos⁡(A−B)−cos⁡(A+B)2\sin A\sin B=\cos \left( A-B \right)-\cos \left( A+B \right).