 Suppose we have a product of powers of sine and cosine. Since the derivative of sine is cosine and the derivative of cosine is negative sine, then removing a factor of sine or cosine gives the possibility of using a u-substitution. For example, suppose we want to find the integral of cosine cubed x dx. So if we remove a factor of cosine, we can rewrite the remainder using the trigonometric identity, cosine squared plus sine squared equals one. And so cosine squared x can be rewritten as, and now if we let u equals sine x, du equals cosine x dx, our integral then becomes, and we can find the antiderivative, then put things back where we found them, u was equal to sine x. Or we might have a power of sine, so we can split off a factor of sine and then rewrite the remainder in terms of cosine. So sine to the fourth x is really sine squared squared, which is one minus cosine squared, which we can expand. And now if we let u equals cosine x, du is negative sine x, and so our integral becomes, which we can evaluate, then put everything back where we found it. Or if we try cosine to the fourth this way, we'd break off a factor of cosine and then try to rewrite the remainder in terms of sine. Now using cosine squared x plus sine squared x equals one gives us cosine x equals square one minus sine squared x square will omit the plus minus for now pending the determination of the domain. Then cosine cubed becomes, and so we have this integral. But we can't evaluate this integral. Fortunately we have another approach, we can reduce a power of sine or cosine using the half angle formulas, cosine squared x equals one half plus one half cosine two x, and then sine squared x equals one half minus one half cosine two x. And these emerge from the trigonometric identities for the double angle plus the Pythagorean identity. So if I want to evaluate cosine to the fourth x dx, well cosine to the fourth x dx is really cosine squared squared. We can use those half angle identities and take our cosine squared and reduce it to cosine of two x, expand. Now notice that this gives us a cosine squared again, and so again we can use our half angle formula a second time to reduce that. And since there are no powers of sine or cosine we can integrate this directly.