 We often encounter integrals of products of sine and cosine. We can transform these expressions using the angle, sum, and difference identities. For example, let's try to rewrite sine 3x cosine 5x as the sum of sines or cosines. So let's pull in those angle sum formulas, and we notice that this product of sine 3x cosine 5x would appear in the formula for sine 3x plus 5x. So let's write that down. This product would also appear in the formula for sine 3x minus 5x. So we'll write that down as well, and we note that if we add them, we can eliminate the term we don't want cosine 3x sine 5x. So we'll add them, and we can simplify the arguments, and we can solve for sine 3x cosine 5x, and we can make one more simplification. Since sine of negative theta is negative sine of theta, we can simplify this sine of negative 2x as minus the sine of 2x. And so we can write our final answer as... So how does this apply to integrals? So remember, there are really only two integration techniques, U-substitutions and integration by parts. Everything else is algebra or trigonometry. So in this case, we can simplify trigonometrically sine 3x cosine 5x as sine 8x minus sine 2x divided by 2. We'll remove that common factor, and we'll integrate the terms. What about a product of cosines? So we note that from our angle sum formulas, the product of cosines appears in the expansions of the cosine of a sum or difference. So cosine 5x cosine 2x would appear in the expansion of cosine 5x plus 2x, and also in the expansion cosine 5x minus 2x. Expanding. And this time, if we add them, we'll eliminate the sine 5x sine 2x terms, and we can solve for cosine 5x cosine 2x. We can remove that factor of 1 half and integrate. And for a product of sines, again, we have our angle sum formulas, and our product of sines appears in the expansion of cosine 2x plus 5x and cosine 2x minus 5x. Now, this time, to eliminate the term we don't want, we'll need to subtract the cosine 2x plus 5x, or multiply the first equation by negative 1 and add the 2, solving for sine 2x sine 5x. And remember, cosine is an even function, so cosine of negative theta is cosine of theta, so we can simplify our expression too. And so our integrand becomes, and we can now do the calculus. So we could write formulas for the conversion of a product of sines or cosines into a sum of sines or cosines, but we won't. Remember, understand concepts don't memorize formulas. These are based on the concept of using the angle sum formulas. Now the key idea is don't begin by memorizing formulas, understand where they come from.