Problem 53 - Precalculus Essentials

Question

Verify each identity. \(\tan 2 \alpha=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}\) IIint: Write \(\tan 2 \alpha\) as \(\tan (\alpha+\alpha)\)

Step-by-Step Solution

Answer

The steps detailed verify the trigonometric identity \(\tan 2\alpha= \frac{2 \tan\alpha}{1 - (\tan\alpha)^2}\).

Step 1: Represent \(\tan 2\alpha\) as \(\tan(\alpha + \alpha)\)

The first step is to express \(\tan 2\alpha\) as \(\tan(\alpha + \alpha)\) because the identity we want to prove involves tangent of twice angle \(\alpha\). So this step simply extends the problem to a form that can be better manipulated.

Step 2: Apply the sum formula for tangent

The formula for the tangent of the sum of two angles is:\(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). We apply this formula here, so \(\tan(\alpha + \alpha) = \frac{\tan\alpha + \tan\alpha}{1 - \tan\alpha \tan\alpha}\). Combine the terms in the numerator then you get \(\frac{2\tan\alpha}{1 - (\tan\alpha)^2}\)

Step 3: Check and rewrite equation

After these transformations, we see that the left-side of the equation turns into \(\frac{2\tan\alpha}{1 - (\tan\alpha)^2}\), which is the same as the right-side of the given identity. So, the transformed and given equations are the same, therefore the identity is verified.

Key Concepts

Double Angle Identities

Double angle identities are essential tools in trigonometry. They allow us to express trigonometric functions of double angles in terms of single angles. Specifically, for sine, cosine, and tangent functions, these identities simplify complex expressions.

The double angle identity for tangent is expressed as \( \tan(2\alpha) = \frac{2 \tan(\alpha)}{1-\tan^2(\alpha)} \). This formula helps in simplifying expressions where the angle is doubled.

Understanding such identities can be extremely helpful when solving problems involving trigonometric expressions, particularly when you need to reduce the power of trigonometric functions or simplify the calculation for computational purposes.

Tangent Function

The tangent function is one of the basic trigonometric functions, alongside sine and cosine. It's defined as the ratio of the opposite side to the adjacent side in a right-angled triangle when considering an angle \( \alpha \). Mathematically, it's expressed as \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \).

Tangent has a periodicity of \( \pi \) (or 180 degrees), and unlike sine and cosine, it has vertical asymptotes where \( \cos(\alpha) = 0 \).

To leverage the tangent function in identities, you often use its relationship with sine and cosine. For instance, remembering its ratio nature can help when transforming more complex identities, just like in the exercise where the double angle identity for tangent was verified.

Sum of Angles Formula

The sum of angles formula is another fundamental tool in trigonometry, especially useful when dealing with expressions involving the sum of two angles. For the tangent function, it is given by:

• \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \).

This formula is crucial when breaking down expressions and reformatting them to solve or prove identities.

In the given exercise, the identity \( \tan(2\alpha) \) was broken down using the concept of the sum \( (\alpha + \alpha) \). By substituting \( \alpha + \alpha \) into the sum of angles formula, the solution neatly showed how the identity is verified. Such understanding allows for easier manipulation of trigonometric equations in both academic problems and real-world applications.