Question

In Problems 57– 80, solve each equation on the interval $0\le \theta \le 2\mathrm{\pi }$

${\sec }^2\theta +\tan \theta =0$

Short answer

The given equation has no real solution.

Step-by-step solution

Step 1. Given Information 

In the given problems we have to solve each equation on the interval $0\le \theta \le 2\mathrm{\pi }$

${\sec }^2\theta +\tan \theta =0$

Step 2. The equation in its present form contains a secant and tangent.  

However, a form of the Pythagorean Identity, ${\sec }^2\theta -{\tan }^2\theta =1$, can be used to transform the equation into an equivalent one containing only tangent.

$$\begin{gathered} 1+{\tan }^2\theta +\tan \theta =0 \\ {\tan }^2\theta +\tan \theta +1=0 \end{gathered}$$

Step 3. We can not find the factor of the given equation.

So this equation does not have any real solution.