# How do you use the sum to product formulas to write the sum or difference #cos(phi+2pi)+cosphi# as a product?

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Using the sum-to-product formulas for cosine, the expression ( \cos(\phi + 2\pi) + \cos(\phi) ) can be written as a product.

The sum-to-product formula for cosine states:

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

In this case, let ( A = \phi + 2\pi ) and ( B = \phi ):

[ A + B = (\phi + 2\pi) + \phi = 2\phi + 2\pi ] [ A - B = (\phi + 2\pi) - \phi = \pi ]

Now, substitute these values into the formula:

[ \cos(\phi + 2\pi) + \cos(\phi) = 2\cos\left(\frac{2\phi + 2\pi}{2}\right)\cos\left(\frac{\pi}{2}\right) ]

[ = 2\cos(\phi + \pi)\cos\left(\frac{\pi}{2}\right) ]

[ = 2(-\sin(\phi))\cdot 0 ]

[ = 0 ]

Therefore, ( \cos(\phi + 2\pi) + \cos(\phi) ) can be written as the product ( 0 ).

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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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