Chapter 7.1, Problem 5CFU

To determine

To find: whose statement is correct.

Answer to Problem 5CFU

Rosalinda

Explanation of Solution

Given information:

Claude’s statement: Since he has tried ten specific values for the variable and all of them worked so the given equation is an identity.

Rosalinda’s statement: Specific values could only be used as counterexamples to prove that an equation is not an identity.

Calculation:

Let’s check the equation $1-\sin x=\cos x$ , is an identity or not.

Put $x=\frac{\pi }{2}$ in the given equation and simplify as shown below.

  $\begin{array}{c}1-\sin \frac{\pi }{2}=\cos \frac{\pi }{2}\\ 1-0=1\\ 1=1\end{array}$

Put $x=\frac{\pi }{6}$ in the given equation and simplify as shown below.

  $\begin{array}{c}1-\sin \frac{\pi }{6}=\cos \frac{\pi }{6}\\ 1-\frac{1}{2}=\frac{\sqrt{3}}{2}\\ \frac{1}{2}\ne \frac{\sqrt{3}}{2}\end{array}$

It is clear that just trying specific values does not prove an identity because there could be other specific values for which it fails. Thus Rosalinda is correct.