Chapter 5.1, Problem 53E

Solution

To find: All solutions to the equation $\tan x{\sin }^{2}x=\tan x$ in the interval $\left[0,2\pi \right)$ .

The solutions to the equation $\tan x{\sin }^{2}x=\tan x$ in the interval $\left[0,2\pi \right)$ is $\left\{0,\pi ,\frac{\pi }{2},\frac{3\pi }{2}\right\}$ .

Given information:

The given trigonometric equation is $\tan x{\sin }^{2}x=\tan x$

Formula used:

The general solution of $\tan x=0$ in the interval $\left[0,2\pi \right)$ is $\left\{0,\pi \right\}$ .

The general solution of $\cos x=0$ in the interval $\left[0,2\pi \right)$ is given by $\left\{\frac{\pi }{2},\frac{3\pi }{2}\right\}$ .

Calculation:

Consider the given equation,

  $\begin{array}{c}\tan x{\sin }^{2}x=\tan x\\ \tan x{\sin }^{2}x-\tan x=0\end{array}$

Take the common factor out.

  $\begin{array}{c}\tan x\left({\sin }^{2}x-1\right)=0\\ \tan x\left(-{\cos }^{2}x\right)=0\end{array}$

On simplify, there are two possibilities,

Either $\tan x=0$ or $-{\cos }^{2}x=0$

For $\tan x=0$ , the general solution in the interval $\left[0,2\pi \right)$ includes,

  $x=0,\pi$

For $\cos x=0$ , the general solution in the interval $\left[0,2\pi \right)$ includes,

  $x=\frac{\pi }{2},\frac{3\pi }{2}$

Thus, the solutions to the equation $\tan x{\sin }^{2}x=\tan x$ in the interval $\left[0,2\pi \right)$ is $\left\{0,\pi ,\frac{\pi }{2},\frac{3\pi }{2}\right\}$ .