Question

Determining Identities Graphically

Graph $f$and $g$in the same viewing rectangle. Do the graphs suggest that the equation $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}$is an identity? Prove your answer

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{x}\mathbf{-}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}\mathbf{,}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}$

Short answer

The expression$f\left(x\right)=g\left(x\right)$ is an identity.

Step-by-step solution

Step 1. Given information

Two functions as-

$\begin{array}{l}f\left(x\right)={\cos }^{2}x-{\sin }^{2}x\\ g\left(x\right)=1-2{\sin }^{2}x\end{array}$

Step 2. Concept used

Create distinct graphs for both functions on the same graph. Both functions are equivalent if both graphs coincide with one other. As a result, combining them into a single equation yields an identity.

Step 3. Calculation

Now, draw graphs of both functions:

Since both graphs coincide each other. Therefore$f\left(x\right)=g\left(x\right)$ is an identity