Chapter 4.3, Problem 79E

To determine

(a)

To show:

The equation ${\left[f\left(x\right)\right]}^2+{\left[f^{\prime }\left(x\right)\right]}^2={\left[f\left(0\right)\right]}^2+{\left[f^{\prime }\left(0\right)\right]}^2$ holds for all $x$.

To determine

(b)

To show:

That $f\left(x\right)=\sin x$ and $f\left(x\right)=\cos x$ satisfy the equation $f^{″}\left(x\right)=-f\left(x\right)$ and deduce that ${\sin }^{2}x+{\cos }^{2}x=1$.