Question

Is the Equation an Identity? Determine whether the given equation is an identity. If the equation is not an identity, find all its solutions.

$e^{{\sin }^{2}x}{e}^{{\cos }^{2}x}=e$

Short answer

$e^{{\sin }^{2}x}{e}^{{\cos }^{2}x}=e$

Step-by-step solution

Step 1. Given information.

$\mathrm{The}\mathrm{Given}\mathrm{equation},{\mathrm{e}}^{{\sin }^2\mathrm{x}}{\mathrm{e}}^{{\cos }^2\mathrm{x}}=\mathrm{e}.$

Step 2. - First consider left hand side and apply exponential identity and proceed.

First consider left hand side and apply exponential identity and proceed.

Given equation is:$e^{{\sin }^{2}x}{e}^{{\cos }^{2}x}=e$

Simplifying L.H.S:

$=e^{{\sin }^{2}x}\cdot e^{{\cos }^{2}x}$

Using identity ${\text{e}}^x{\text{e}}^y=e^{x+y}$

$e^{{\sin }^{2}x}\cdot e^{{\cos }^{2}x}=e^{{\sin }^{2}x+{\cos }^{2}x}$

$\begin{array}{l}\because {\sin }^{2}x+{\cos }^{2}x=1\\ e^{{\sin }^{2}x}\cdot e^{{\cos }^{2}x}=e\end{array}$

we get,
L.H.S = e