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

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

Short answer

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

Step-by-step solution

Step 1. Given information

Two functions as-

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

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