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Chapter 5: Problem 119

Verify the identity. $$\tan \frac{u}{2}=\csc u-\cot u$$

Short Answer

Expert verified
The identity \(\tan \frac{u}{2} = \csc u - \cot u\) has been successfully verified, by expressing all terms in sine and cosine, simplifying the RHS, using the double-angle identity for sine, and cancelling common terms.

Step by step solution

01

Express in terms of sine and cosine

Transform all trigonometric functions into sine and cosine. Hence, \(\tan \frac{u}{2} = \frac{\sin \frac{u}{2}}{\cos \frac{u}{2}}\) and \(\csc u - \cot u = \frac{1}{\sin u} - \frac{\cos u}{\sin u}\). Now it becomes simpler because we have only sine and cosine functions in the equation.
02

Simplify the RHS

The right-hand side of the equation \(\frac{1}{\sin u} - \frac{\cos u}{\sin u}\) can be further simplified, as both fractions have the same denominator. It can be re-written as \(\frac{1-\cos u}{\sin u}\). Now, use the Pythagorean identity \(1-\cos^2 u = \sin^2 u\) which further transforms the RHS to \(\frac{\sin^2 u}{\sin u} = \sin u\).
03

Use double-angle identity for sine

The double-angle identity for sine is given by \(\sin u = 2 \sin \frac{u}{2} \cos \frac{u}{2}\). Thus, replace \(\sin u\) with \(2 \sin \frac{u}{2} \cos \frac{u}{2}\) in the expression obtained in the previous step.
04

Cancel common terms and verify

In the expression \(2 \sin \frac{u}{2} \cos \frac{u}{2}\), cancel \(\sin \frac{u}{2}\) from both sides. You will now find that both sides of the initial equation are identically equal to \(\frac{\sin \frac{u}{2}}{\cos \frac{u}{2}}\), thereby verifying the given identity.

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