Look at the Taylor series for e^x, and see what you get for e^(i * theta). The terms with i raised to an even power are real, those with i raised to an odd powe

Look at the Taylor series for e^x, and see what you get for e^(i * theta). The terms with i raised to an even power are real, those with i raised to an odd power are imaginary. Group the odds together and the evens together, and compare with what the Taylor series for sin x and cos x give you for sin theta and cos theta. (I'm being a bad person here, ignoring convergence issues, but I want to say that all three are absolutely convergent.)