I need some help in these exercices:
1.1 - Prove that tan(x+y)= [tan(x)+tan(y)] / [1−tan(x).tan(y)] and then use this to prove that arctg(x) + arctg(y) = arctg ( [x+y] / [1-xy] ) indicating any possible restriction of arguments.
1.2 - Conclude that: pi / 4 = arctg (1 / 2) + arctg (1 / 3) and pi / 4 = 4.arctg (1 / 5) - arctg (1 / 239)
1.3 - Use this last equation and Taylor's
polynomials of arctg(x) to prove that: pi = 3.14159...
2.1 - Considering a function such that f''(x) + f(x) = 0 for all x belonging to lR, and that f(0) = 0, f'(0) = 0. Prove that all derivatives exist.
2.2 - Calculate Taylor's polynomial of this fucntion at the point 0 and the respective rest.
2.3 - Conclude that any function satisfying these conditions is necessarily null.
I hope someone can help me!
Thank you very much