In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power series:$$\sin(x)=\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$$ and $$\cos(x)=\sum_{k=0}^\infty\frac{(-1)^kx^{2k}}{(2k)!}$$
However, if I now take the squares of these values I get a really messy result that I can't simplify to 1.
Could anyone give me a hint on how to deal with this question?
Best Answer
Hint: Differentiate $\cos^2x+\sin^2x$, and notice that it is $2\cos x\cos'x+2\sin x\sin'x$. Evaluate the expressions of $\sin'x$ and $\cos'x$, and notice that they are the same as those of $\cos x$ and $-\sin x$ respectively. Then you get $\big(\cos^2x+\sin^2x\big)'=0\iff\cos^2x+\sin^2x=$ constant. How do we guess which constant exactly ? By computing $\sin^20+\cos^20$.