Trig Identities. Consider the indefinite integral

Question

where 'a' is a constant

 Evaluate the integral by making the substitution

 Use the cosine addition identity to rewrite the integrand and then evaluate the integral.

 What identity do you get when you compare the results from parts (a) and (b)?

Answer 1

Using u-substitution with u = (Θ+α) and du = dΘ we get

∫cosudu = sinu + C = sin(Θ+α) + C

If instead we use the sum formula for cosine we get

∫cosΘcosα - sinΘsinα dΘ = cosαsinΘ + sinαcosΘ + C = sin(Θ+α) + C , which is the sine addition identity.