Express product of a trigonometric function as a sum or difference

How do I find sin(8x)sin(4x) as a sum or difference of trigonometric functions? I know the sum and difference identities of sin, cos, and tan, but I do not know how to convert from a product of trigonometric functions to a sum of trigonometric functions. Do I use the sin or cos sum and difference identities? I have looked at Product-to-sum trigonometry identity but it did not help.

You will notice that the product formulas for $\sin,\cos$ are not the easiest to retain in memory for any length of time. However, they can be easily derived in a couple of minutes if they are forgotten at a crucial stage, such as during an exam.

They all come from the sum and difference formulas.

To derive the formula for $\sin(A)\sin(B)$ for example, recall that you usually see such terms in the sum formulas for the cosine function.

$$ \cos(A+B)=\cos A\cos B-\sin A\sin B $$

$$ \cos(A-B)=\cos A\cos B+\sin A\sin B $$

We can get an equation with $\sin A\sin B$ on one side by subtracting the first equation from the second to obtain

$$ \cos(A-B)-\cos(A+B)=2\sin A\sin B $$

So we quickly derive

$$ \sin A\sin B =\frac{1}{2}[\cos(A-B)-\cos(A+B)]$$

Using the same two identities we can obtain an identity for $\cos A\cos B$ by adding the two identities.

To obtain the identities for the product of a sine and cosine, use the addition and subtraction identities for the $\sin(A\pm B)$.