Question

Determining Identities Graphically

Graph $\mathit{f}$and $\mathit{g}$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

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

Short answer

The expression$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ is not an identity.

Step-by-step solution

Step 1. Given information

Two functions as-

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

Step 2. Concept used

On the same graph, draw distinct graphs for both functions. Both functions are equivalent if both graphs are parallel to each other. As a result, by combining them into a single equation, an identity is established.

Step 3. Calculation

Now, draw graphs of both functions:

Since both graphs are not coincide each other. Therefore$f\left(x\right)=g\left(x\right)$ is not an identity