Question

Find all exact solutions of the equation on the interval [0, 2π).

cosxtanx=cosx

Answer 1

`\displaystyle x=\left\{(\pi)/(4), (5\pi)/(4)\right\}`

Explanation:

We want to solve the equation:

`\cos(x)\tan(x)=\cos(x)`

On the interval [0, 2π).

First, we can subtract cos(x) from both sides:

`\cos(x)\tan(x)-\cos(x)=0`

Factor:

`\cos(x)\left(\tan(x)-1\right)=0`

Zero Product Property:

`\cos(x)=0\text{ or } \tan(x)-1=0`

Solve for each case:

`\cos(x)=0\text{ or }\tan(x)=1`

Using the unit circle:

`\displaystyle x=\left\{(\pi)/(4), (\pi)/(2), (5\pi)/(4), (3\pi)/(2)\right\}`

However, since tangent isn't defined for π/2 and 3π/2, we remove them from our solutions. Hence:

`\displaystyle x=\left\{(\pi)/(4), (5\pi)/(4)\right\}`