Problem 66 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \tan\left(\dfrac{\pi}{4} - \theta\right) = \dfrac{1 - \tan \theta}{1 + \tan \theta} \)

Step-by-Step Solution

Answer

The identity is successfully proved by applying the addition/subtraction formulas in trigonometry and simplifying the equation

Step 1 Express the tan function

Use the identity \( \tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B} \) to express the left hand side (LHS). So, \( \tan\left(\dfrac{\pi}{4} - \theta\right) = \dfrac{\tan (\pi /4 - \theta)}{1 + \tan (\pi/4) \tan (-\theta)} \).

Step 2 Express tan as sin/cos

Express \(\tan \theta\) as \( \frac{\sin \theta}{\cos \theta}\) and \( \tan \frac{\pi}{4} \) as 1 for further simplification so \( \frac{\tan (\pi /4) - \tan (\theta)}{1 + \tan (\pi/4) \tan (-\theta)} \) will be \( \frac{1 - \frac{\sin \theta}{\cos \theta}}{1 + 1.\frac{\sin \theta}{\cos \theta}} \)

Step 3 Simplify the Equation

After multiplying by cos θ in the numerator and denominator to remove the fraction, the equation simplifies to \( \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \)

Key Concepts

Tangent Function

The tangent function, often abbreviated as "tan," is a fundamental part of trigonometry. It is one of the six common trigonometric functions and is defined as the ratio of the sine to the cosine of a given angle. Mathematically, it is expressed as:

• \( an heta = \frac{\sin \theta}{\cos \theta} \)

This function is particularly useful because it provides a simple way to relate the angles in a right triangle to the lengths of the sides.
In practical applications, tangent is widely used to explore angles in different contexts, from physics to engineering. The basic properties of the tangent function make it an essential part of proving trigonometric identities.
In our exercise, the tangent function undergoes a transformation as part of an angle subtraction formula, demonstrating its importance in complex calculations as well as showcasing its versatility.

Angle Subtraction Formula

The angle subtraction formula for tangent provides a specific way to express the tangent of a difference between two angles. It's a critical identity used to solve many trigonometric equations.
This formula is given by:

• \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

The angle subtraction formula allows for the simplification and solution of trigonometric expressions involving a difference of angles such as \( \pi/4 - \theta \).
By using such formulas, we can seamlessly break down complex problems into more manageable parts. This is especially vital in the proofs where you substitute parts of the identity and come closer to a simplified form.
Understanding the angle subtraction formula is key to learning more advanced trigonometric concepts as it lays the groundwork for solving and proving many identities.

Sine and Cosine Ratios

Sine and cosine are two of the most fundamental trigonometric ratios in mathematics. These functions relate the angles of a right triangle to the ratios of two of its sides.

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Sine and cosine are periodic, with a period of \(2\pi\), and they oscillate between -1 and 1. These attributes make them highly useful in modeling wave-like phenomena.
In the context of the given exercise, expressing tangent in terms of sine and cosine plays a crucial role. When \( \tan \theta \) is written as \( \frac{\sin \theta}{\cos \theta} \), it enables us to simplify identities and to eliminate the tangent in favor of these more foundational functions.
This simplification allows further reduction into a more tractable form, such as the expression \( \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \).
Mastery of sine and cosine, along with their relationship with tangent, deepens one’s understanding of the intricacies involved in trigonometric identities and aids in problem solving effectively.