In the XVI century astronomers derived the following identities, which were called Prosthaphaere- sis formulas (from the Greek Prosthesis = Addition and Apharesis = Subtraction) (1) sin a sin 3 = cos(a - 3)-cos(a + 3) 2 (2) (3) cos a cos 3 sin a cos 3 = = cos(a-3) cos(a + B) 2 sin(a + 3) + sin(a - b) 2 (4) cos a sin sin(a + 3) - sin(a B) 2 You can find a purely geometric proof here. Here, I want you to prove them using algebraic means, namely exploiting Euler's formula and the algebra of complex numbers. Reflect on the two proofs (geometric and algebraic) and comment on which one you find easier. Based on the formulas above, if f(t) and g(t) are two signals with a strong spectral peak at the same frequency w. what can you say about the spectrum of the signal h(t) = f(t)g(t)?