[33 marks in total] The theory of differential equations implies that there are unique functions f, g : R- R such that f'(x) = g(x) for all r E R; (1) g'(x) = -f(x) for all r E R; (2) f(0) = 0; (3) g(0) = 1. (4) The goal is to prove the following three formulae for all a, r E R: (f (x)) 2 + (9(2))2 = 1; (5) f(x + a) = f(a)g(x) + g(a)f(x); (6) g(x + a) = g(a)g(x) - f(a)f(x). (7 ) Daniel suggests the following proof. Step 1 We prove the following lemma: Lemma 1. Let 0 be a differentiable function on R. If O'(x) = 0 for every r E R, then 0 is a constant on R. Suppose that O is not a constant. Then there exist 21, 22 E R such that 0(x1) # 0(x2). WLOG, assume that 21 0, the Intermediate Value Theorem implies that p(x) = 0 for some r E (0, I*), contradicting Step 8; if a*