Problem 53 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \begin{aligned} &\tan 2 \alpha=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}\\\ &\text {Hint: Write } \tan 2 \alpha \text { as } \tan (\alpha+\alpha) \end{aligned} $$

Step-by-Step Solution

Answer

After expressing \( \tan 2 \alpha \) as \( \tan(\alpha + \alpha) \) and applying the sum of angles identity, we can simplify the equation to \( \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \). Therefore, the original given identity \( \tan 2 \alpha = \frac{2 \tan \alpha}{1-\tan^2 \alpha} \) is verified.

Step 1: Express the left-hand side of the equation

Express \( \tan 2 \alpha \) as \( \tan(\alpha + \alpha)\). This gives us \( \tan(\alpha + \alpha) \).

Step 2: Apply the sum of angle identity

The identity of tangent of sum of two angles is used, which is \( \tan u + v = \frac{\tan u + \tan v}{1 - \tan u\tan v}\). Replacing \( u \) and \( v \) with \( \alpha \), the equation becomes \( \frac{\tan \alpha + \tan \alpha}{1 - \tan \alpha\tan \alpha} \).

Step 3: Simplify the expression

This equation simplifies to \( \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \).

Key Concepts

Sum and Difference Identities

Sum and Difference Identities are fundamental tools in trigonometry helping to simplify complex trigonometric expressions. They enable you to express trigonometric functions of summed or subtracted angles in terms of trigonometric functions of the individual angles.
These identities apply to sine, cosine, and tangent functions.
To illustrate:

• The sine of the sum of two angles: \(\sin(u+v) = \sin u \cos v + \cos u \sin v\)
• The cosine of the difference of two angles: \(\cos(u-v) = \cos u \cos v + \sin u \sin v\)
• Similarly, the "sum" identity for tangent: \(\tan(u+v) = \frac{\tan u + \tan v}{1 - \tan u \tan v}\)

These identities not only simplify calculations but also play a crucial role in verifying trigonometric expressions.
When you encounter exercise tasks dealing with sum or difference, remember these identities to unlock the solutions effectively.

Tangent Function

The tangent function, often abbreviated as \(\tan\), is one of the fundamental trigonometric functions. It relates the angles of a right triangle to the ratio of two of its sides: the opposite side over the adjacent side.
More formally, for any angle \(\theta\), the tangent is defined as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
This function helps in calculating various properties associated with angles, especially in periodic phenomena.
The tangent function is unique because it has a period of \(\pi\), meaning it repeats its values every \(\pi\) radians.

• Tangent is undefined where \(\cos(\theta) = 0\), causing vertical asymptotes in its graph.
• This graph has an "S" like shape, periodic, and exhibits reflection symmetry.

When solving problems like the provided exercise, understanding the nature of tangent helps in applying identities and simplifying expressions smoothly.
Thus, mastering tangent function properties aids in tackling a wide range of trigonometric challenges.

Angle Sum Identity

The Angle Sum Identity allows you to find the trigonometric functions of an angle expressed as the sum of two other angles. It is an important identity that makes solving trigonometric equations easier.
For tangent, the Angle Sum Identity is particularly useful: \(\tan(u+v) = \frac{\tan u + \tan v}{1-\tan u \tan v}\).
This identity helps break down complex angles into simpler parts that are easier to work with.
In our particular exercise, using the angle sum identity for tangent, we expressed \(\tan(2\alpha)\) as \(\tan(\alpha + \alpha)\), simplifying it to \(\frac{2\tan(\alpha)}{1-\tan^2(\alpha)}\).

• It shows how a seemingly complex double angle can be represented using a known identity.
• This not only confirms the correctness of the identity but also enhances understanding of angle properties.

By leveraging the angle sum identities, the manipulation and simplification of trigonometric expressions become more accessible, making them an invaluable asset for students tackling such problems.