# How do you use sum to product formulas to write the sum or difference #sin(x+pi/4)-sin(x-pi/4)# as a product?

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To express ( \sin(x + \frac{\pi}{4}) - \sin(x - \frac{\pi}{4}) ) as a product, we utilize the sum-to-product formula for sine, which states:

[ \sin(A) - \sin(B) = 2\sin\left(\frac{A-B}{2}\right)\cos\left(\frac{A+B}{2}\right) ]

Applying this formula to the given expression, we have:

[ \sin(x + \frac{\pi}{4}) - \sin(x - \frac{\pi}{4}) = 2\sin\left(\frac{(x + \frac{\pi}{4}) - (x - \frac{\pi}{4})}{2}\right)\cos\left(\frac{(x + \frac{\pi}{4}) + (x - \frac{\pi}{4})}{2}\right) ]

[ = 2\sin\left(\frac{\frac{\pi}{2}}{2}\right)\cos\left(\frac{2x}{2}\right) ]

[ = 2\sin\left(\frac{\pi}{4}\right)\cos(x) ]

So, ( \sin(x + \frac{\pi}{4}) - \sin(x - \frac{\pi}{4}) ) can be expressed as ( 2\sin\left(\frac{\pi}{4}\right)\cos(x) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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