Problem 60 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \tan \frac{x}{2}=\frac{\sin x}{1+\cos x} $$

Step-by-Step Solution

Answer

The identity is verified using trigonometric manipulation.

Step 1: Use Double Angle Formulas

To verify the identity \( \tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} \), start by recalling the double angle identities. Specifically, for the tangent of half an angle: \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \). We will work to transform this expression to match the right side of the identity given.

Step 2: Multiply Numerator and Denominator by Conjugate

To simplify \( \frac{1 - \cos x}{\sin x} \), multiply both the numerator and the denominator by \( 1 + \cos x \), its conjugate. This results in:\[\frac{(1 - \cos x)(1 + \cos x)}{\sin x (1 + \cos x)}\]The purpose of this is to use the difference of squares to simplify the numerator.

Step 3: Simplify the Numerator

Simplify the expression \((1 - \cos x)(1 + \cos x)\), using the difference of squares formula:\[(1 - \cos x)(1 + \cos x) = 1 - \cos^2 x = \sin^2 x\]This simplification results in the expression:\[\frac{\sin^2 x}{\sin x (1 + \cos x)}\]

Step 4: Simplify the Resulting Expression

Cancel out \(\sin x\) in the numerator and the denominator:\[\frac{\sin^2 x}{\sin x (1 + \cos x)} = \frac{\sin x}{1 + \cos x}\]

Step 5: Conclude Verification

We have reached \( \frac{\sin x}{1 + \cos x} \), which matches the right-hand side of the given identity. Therefore, the identity is verified.

Key Concepts

Double Angle Identities

Double angle identities are an essential aspect of trigonometry. They allow us to express trigonometric functions of double angles in terms of single angles. These identities are particularly useful when working with transformations and verifications of trigonometric identities. In the context of verifying identities, we often need to manipulate expressions into equivalent forms, which is where double angle formulas can be extremely helpful.

For example, the double angle formula for tangent indicates that:

• \( \tan(2x) = \frac{2 \tan x}{1 - \tan^2 x} \)

Yet, in our specific problem, we're exploring a transformation stemming from half angles, not double, but the logic to simplify remains the same—use trigonometric identities to simplify or transform expressions.

Knowing the identities for sine, cosine, and tangent at double or half angles equips you with a toolbox for handling complex trigonometric problems effectively.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables. They include several key sets of formulas such as Pythagorean identities, reciprocal identities, and of course, angle identities (double angle and half-angle formulas being subsets of these).

These identities are useful for simplifying expressions and solving equations where trigonometric functions are involved. For our exercise, using the identity \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \) is crucial. This particular identity helps us transform and verify the expression you’re given into an equivalent form. The interplay of different identities is a powerful technique in trigonometry that assists in simplifying and proving equations.

Half-Angle Formulas

Half-angle formulas are another important set of identities in trigonometry. They involve trig functions and express angles as half of another angle. These formulas are derived from the double angle formulas and simplify a variety of trigonometric expressions.

In our example problem, the focus is on the formula for tangent:

• \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \)

This identity is central to the solution. In resolving the equation \( \tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} \), we began with the left side formula \( \frac{1 - \cos x}{\sin x} \) and shifted it through strategic manipulations to match the right side.

The half-angle formulas, like the other trig identities, are indispensable for verifying and reducing complex trigonometric expressions, making them easier to work with.