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Recently in class, my professor went over a relationship that exists between trigonometric functions, T(x), and their complementary functions. That is:

∫ T(x)dx = W(x) + C

∫ coT(x)dx = -co[W(x)] + C

Without really providing a proof, we were told this relationship. I've plugged in numbers and tried graphing the scenario and it works from what I've done so far. If anyone could provide me a formal proof through any source, that would be great since I need a bit more convincing.

Also, is a limitation to this property that both W(x) and T(x) must be a combination trigonometric functions themselves? Since when trying to solve ∫cscxdx using the answer from ∫secdx, an answer cannot be found. Are there any other limitations?

Also, regarding the identities:

sinAcosB = 1/2 [sin(A-B) + sin(A+B)

sinAsinB = 1/2 [cos(A-B) - cos(A+B)]

cosAcosB = 1/2[cos(A-B) + cos(A+B)]

Is there a formal proof readily accessible for these identities?

Thanks!