Chapter 5.5, Problem 109E

To determine

To verify: The given identity.

Explanation of Solution

Given information:

The following identity:

  $\csc \frac{u}{2}=\pm \sqrt{\frac{2\csc u}{\csc u-\cot u}}$

Formula used:

The following half angle formula:

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

The following Pythagorean relation:

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

Proof:

The given identity is verified as follows:

Working with the left side,

  $\begin{array}{l}\csc \frac{u}{2}=\frac{1}{\sin \frac{u}{2}}\text{ [Pythagorean relation]}\\ =\pm \frac{1}{\sqrt{\frac{1-\cos u}{2}}}\text{ [Half angle formula]}\\ =\pm \frac{\sqrt{2}}{\sqrt{1-\cos u}}\text{ [Multiply numerator and denominator by }\sqrt{2}]\\ =\pm \frac{\sqrt{\frac{2}{\sin u}}}{\sqrt{\frac{1-\cos u}{\sin u}}}\text{ [Divide numerator and denominator by }\sin u]\\ =\pm \frac{\sqrt{\frac{2}{\sin u}}}{\sqrt{\frac{1}{\sin u}-\frac{\cos u}{\sin u}}}\text{ [Seperate fractions]}\\ =\pm \sqrt{\frac{2\csc u}{\csc u-\cot u}}\text{ [Pythagorean relations]}\end{array}$

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 ${\text{Y}}_1=\csc \frac{u}{2}$ , ${\text{Y}}_2=\sqrt{\frac{2\csc u}{\csc u-\cot u}}$ and ${\text{Y}}_3=-\sqrt{\frac{2\csc u}{\csc u-\cot u}}$ which are 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.