i need help w/ Pre Cal hw?

Question

1) Prove the identities for sin(u+v) and cos(u+v) . By using Euler identities.


2) To prove the identities for sin(u-v) and cos(u-v), start with e^i(u-v)= e^iu e^-iv and repeat.


3) (Euler identities)

e^ix = cos(x) + i sin(x) = cis(x)
e^-ix= cos(x) -i sin (x) = cis(-x)

a) Use the fact that cos (x) and sin(x) are periodic with a period of 2 pie. Recall that a fuction f (x) is periodic w/ period T if f(x + T) = f(x). The exponential form of a complex number can thus b defined as z=re^10 , where r and 0 are the modulus and argument of z, respectively.

b) Treat the Euler identities as a 2 times 2 system of linear equations in sin (x) and cos (x) and solve the system. That is, find sin (x) and cos (x) in terms of e^ix and e^-ix. Verify that these representations of sine and cosine are purely real by taking their complex conjugates.

Answer 1

Eulers identity

e^(i*theta) = cos(theta) + i*sin(theta)
(where i is equal to the square root of -1

For the early parts you are going to have to use
properties of exponents such as

a^(x+y) = a^(x) *a^(y)