Problem 155 - Precalculus

Question

For the following exercises, prove the identities. $$ \tan (2 x)=\frac{2 \sin x \cos x}{2 \cos ^{2} x-1} $$

Step-by-Step Solution

Answer

The identity is proved by substituting the double angle identities for sine and cosine.

Step 1: Express the Left-Side Using Double Angle Formula

The identity we need to prove is \( \tan(2x) = \frac{2\sin x \cos x}{2\cos^2 x - 1} \). We know that \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \). Also, recall the double angle identity for the sine function: \( \sin(2x) = 2\sin x \cos x \). Substitute this into the tangent formula to rewrite the numerator.

Step 2: Express the Cosine of Double Angle

Now, we need to find \( \cos(2x) \) using the double angle identity. The identity is \( \cos(2x) = 2\cos^2 x - 1 \). Substitute this into the tangent formula as the denominator.

Step 3: Substitute and Simplify the Expression

Substitute the double angle expressions into the formula: \( \tan(2x) = \frac{2\sin x \cos x}{2\cos^2 x - 1} \). This matches the right-hand side of the initial identity you are asked to prove, thus confirming the identity.

Key Concepts

Double Angle Formulas

The double angle formulas are crucial tools in trigonometry that help simplify expressions involving trigonometric functions evaluated at double angles. These formulas can transform calculations involving complicated expressions into simpler forms, making them easier to work with. Specifically, when dealing with

• Sine Double Angle Formula: The formula is given by \( \sin(2x) = 2\sin x \cos x \). This formula allows you to express the sine of double an angle in terms of the sine and cosine of the original angle \(x\).
• Cosine Double Angle Formula: One of its forms is \( \cos(2x) = 2\cos^2 x - 1 \). This identity expresses the cosine of double an angle using only the cosine of the angle.

These formulas are widely used for proving identities, simplifying trigonometric expressions, and solving equations. They essentially allow the transformation of complex expressions into potentially easier ones to handle.
Understanding and applying them correctly is fundamental when dealing with identities like the one given in the exercise.

Tangent Function

The tangent function, often denoted as \( \tan \), is a fundamental trigonometric function that relates the sine and cosine functions. It plays a pivotal role in trigonometry, especially when working with identities and transformations. The basic definition of the tangent function is given by

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

This definition reveals how the tangent function is derived from the quotient of sine and cosine. In trigonometric identities involving the tangent of double angles, we often utilize the double angle formulas for both sine and cosine.Consider the identity for the tangent of a double angle:

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

This means you can express \( \tan(2x) \) in terms of \( \sin(2x) \) and \( \cos(2x) \). The given exercise is an excellent example of demonstrating how to manipulate this relation to prove a more complex trigonometric identity.
Understanding these relationships allows for more manageable calculations and simplifications when proving identities or working with trigonometric functions in problems.

Sine and Cosine Functions

Sine and cosine are the foundational trigonometric functions that describe the relationship between the angles and sides of a right triangle. They are essential in forming and understanding other trigonometric functions and identities. The sine function \( \sin(x) \) represents the opposite side over the hypotenuse in a right triangle for an angle \(x\), while the cosine function \( \cos(x) \) represents the adjacent side over the hypotenuse.

• Sine Function: \( \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \) in a right triangle.
• Cosine Function: \( \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \) in a right triangle.

These functions are fundamental for defining other trigonometric identities and functions such as tangent, secant, and others.
In the context of the exercise, the sine and cosine functions are further utilized in their double angle forms to establish the proof for the tangent identity. Recognizing that these two functions serve as the basis for the tangent expression emphasizes their importance not just in solving equations, but in proving identities as well. Moreover, being familiar with their properties aids significantly in simplifying various trigonometric expressions.