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Chapter 5: Problem 56

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\cos \left(\theta+\frac{\pi}{2}\right)-\cos \left(\theta-\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The product equivalent of the given sum of cosines is \(-2\sin(\theta)\)

Step by step solution

01

Identify the variables a and b

Looking at the given sum, variables 'a' and 'b' can be assigned as follows: 'a' is equal to \(\theta+\frac{\pi}{2}\) and 'b' is equal to \(\theta-\frac{\pi}{2}\)
02

Apply the sum-to-product formula for cosine

Substitute 'a' and 'b' into the sum-to-product formula: \(-2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)\) to get \(-2\sin\left(\frac{(\theta+\frac{\pi}{2})+(\theta-\frac{\pi}{2})}{2}\right)\sin\left(\frac{(\theta+\frac{\pi}{2})-(\theta-\frac{\pi}{2})}{2}\right)\)
03

Simplify the resulting expressions

Simplify inside the sine functions to get \(-2\sin\left(\frac{2\theta}{2}\right)\sin\left(\frac{\pi}{2}\right)\), which simplifies to \(-2\sin(\theta)\sin\left(\frac{\pi}{2}\right)\). Do note that \(\sin\left(\frac{\pi}{2}\right)\) is equal to 1
04

Final Simplification

Finally, simplifying the expression yields \(-2\sin(\theta)\)

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