Proving $\sin^2(x) + \cos^2(x) =1$ using calculus Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: 
Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for all $x$. If $s(0)=0$ and $c(0)=1$, prove that $s^2(x) + c^2(x) =1$.
I tried using integrals and derivatives but could only show that the derivative of  $s'(x)+c'(x)$ is $s(x) + c(x)$ but I think that was completely wrong way to think about it. I didn't read too well up on the mean value theorem and Rolle's theorem and perhaps the proof uses those theorems. Anyway could any one show me proof of this? Thanks. 

Take the derivative of $s^2+c^2$ and you'll see it's zero and is therefore a constant function. With the initial conditions, it must be one.
