Chapter 5.5, Problem 102E

To determine

To verify: The given identity.

Explanation of Solution

Given information:

The following identity:

  $\tan \frac{u}{2}=\csc u-\cot u$

Formula used:

The following half angle formula:

  $\tan \frac{u}{2}=\frac{1-\cos u}{\sin u}$

The following trigonometric relations:

  $\begin{array}{l}\frac{1}{\sin u}=\csc u\\ \frac{\cos u}{\sin u}=\cot u\end{array}$

Proof:

The given identity is verified as follows:

Working with the left side,

  $\begin{array}{l}\tan \frac{u}{2}=\frac{1-\cos u}{\sin u}\text{ [Half angle formula]}\\ =\frac{1}{\sin u}-\frac{\cos u}{\sin u}\text{ [Seperate fractions]}\\ =\csc u-\cot u\text{ [Pythagorean relation]}\\ \end{array}$

Since both sides are equal, the identity is verified

The identity is verified graphically as:

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{Y}}_1=\tan \frac{u}{2}$ and ${\text{Y}}_2=\csc u-\cot u$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

Since both the curves coincide, the identity is verified.