Those aren't simple proofs...

here you have a link of a math forum from my country with those proofs (it's in spanish :S, but you can understand if you only read the operations)

Basically, getting 1st and the 3rd and using parity and imparity of cos and sin you can demostrate the others

Funciones trigonométricas de la suma y diferencia de ángulos - Foro fmat.cl

I hope it can help you