Addition formulae proof for sin(x + y) and cos(x + y)

[maximus101]

For a fixed $$y$$ $$\in$$ $$R$$ , if
$$f(x)$$ $$=$$ $$[sin(x+y)]$$$$-(sin x)$$ $$(cos y)$$ − $$(cos x)$$ $$(sin y)$$and we let $$E(x) =$$ $$[f(x)]^2$$+$$[f'(x)]^2$$.

How do we prove the addition formulae for $$sin(x + y)$$and $$cos(x + y)$$ by applying the Mean Value Theorem to $$E$$

 

[LCKurtz]

For a fixed $$y$$ $$\in$$ $$R$$ , if
$$f(x)$$ $$=$$ $$[sin(x+y)]$$$$-(sin x)$$ $$(cos y)$$ − $$(cos x)$$ $$(sin y)$$and we let $$E(x) =$$ $$[f(x)]^2$$+$$[f'(x)]^2$$.

How do we prove the addition formulae for $$sin(x + y)$$and $$cos(x + y)$$ by applying the Mean Value Theorem to $$E$$

Well, if you were to show E(x) ≡ 0 that would show both f and f' are identically 0.