Question

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

$\tan \theta =\cot \theta$

Short answer

The solution set of the given equation is$\left\{\frac{\pi }{4},\frac{3\pi }{4},\frac{5\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}\right\}$

Step-by-step solution

Step 1. Given Information 

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

$\tan \theta =\cot \theta$

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

Divide by $\cot \theta$on both side

$$\begin{gathered} \frac{\tan \theta }{\cot \theta }=\frac{\cot \theta }{\cot \theta } \\ \frac{\tan \theta }{\cot \theta }=1 \end{gathered}$$

We know that $\tan \theta =\frac{1}{\cot \theta }$

So

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

Step 3. Taking square root on both side

$$\begin{gathered} \tan \theta =\pm \sqrt{1} \\ \tan \theta =\pm 1 \\ \tan \theta =1\mathrm{or}\tan \theta =-1 \end{gathered}$$

Step 4. Solving each equation in the interval [0,2π], we obtain

$\theta =\frac{\pi }{4},\frac{5\pi }{4}\mathrm{or}\mathrm{\theta }=\frac{3\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}$

The solution set is$\left\{\frac{\pi }{4},\frac{3\pi }{4},\frac{5\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}\right\}$