Chapter 4.8, Problem 23ES

(a) Use the Constant Difference Theorem (4.8.3) to show that if $f^{\prime }\left(x\right)=g^{\prime }\left(x\right)$ for all $x$ in the interval $\left(-\infty ,+\infty \right),$ and if $f$ and $g$ have the same value at some point $x_0,$ then $f\left(x\right)=g\left(x\right)$ for all $x$ in $\left(-\infty ,+\infty \right)$ .

(b) Use the result in part (a) to confirm the trigonometric identity ${\sin }^{2}x+{\cos }^{2}x=1$ .