How on earth do I square the summation?

Question

I'm supposed to show that (C(x))² + (S(x))² = 1

C(x) = ∑ (from k = 0 to infinity) { [ (-1)^k] / (2k)! } x^(2k)

S(x) = ∑ (from k = 0 to infinity) { [ (-1)^k] / (2k+1)! } x^(2k+1)

I'm not allowed to just say C(x) = cos x and S(x) = sin x so cos²x + sin²x = 1

I was given a hint to let f(x) = (C(x))² + (S(x))² and I have already proven that f ' (x) = 0, so I know that f(x) is equal to some constant, but how do I show that constant is 1?

Answer 1

Since you know that f(x) is constant you can let x = anything. Let x = 0. Your summations become rather simple then :-)