Problem 44 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \tan \left(\frac{\pi}{4}-\theta\right)=\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta} $$

Step-by-Step Solution

Answer

By applying trigonometric difference and quotient identities, we have verified that the given identity holds true: \( \tan \left(\frac{\pi}{4}-\theta\right)=\frac{\cos \theta-\sin \theta}{\cos\theta+\sin \theta}.\)

Step 1: Initialize with given identity

We start with the given identity: \( \tan \left(\frac{\pi}{4}-\theta\right)=\frac{\cos \theta-\sin \theta}{\cos\theta+\sin \theta} \)

Step 2: Start simplification

Using trigonometric difference identity to simplify \(\tan \left(\frac{\pi}{4}-\theta\right)\), it becomes \[\tan \left(\frac{\pi}{4}-\theta\right)=\frac{\tan \left(\frac{\pi}{4}\right)- \tan \left(\theta\right)}{1+ \tan \left(\frac{\pi}{4}\right) \cdot \tan \left(\theta\right)}\]Given that tan(\(\frac{\pi}{4}\)) = 1, we simplify further to \[\frac{1-\tan \theta}{1+ \tan \theta}\]

Step 3: Use quotient identity

Next, we use the quotient identity \(\tan \theta= \frac{\sin \theta}{\cos \theta}\) to replace \(\tan \theta\), it becomes \[\frac{1-\frac{\sin \theta}{\cos \theta}}{1+\frac{\sin \theta}{\cos \theta}}\]Simplify the above expression to obtain \[\frac{\cos \theta- \sin \theta}{\cos \theta+ \sin \theta}\]

Key Concepts

Sum and Difference Formulas

In trigonometry, sum and difference formulas are essential tools for simplifying expressions and solving equations. The sum and difference formulas for tangent are particularly useful in this scenario.

The tangent difference identity is given by:

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

This formula helps to find the tangent of the difference of two angles.

In the exercise, we use this identity to simplify \( \tan \left(\frac{\pi}{4} - \theta\right) \). Recognizing that \( \tan \left(\frac{\pi}{4}\right) = 1 \), we substitute it into the formula:

• \( \tan \left(\frac{\pi}{4} - \theta\right) = \frac{1 - \tan \theta}{1 + \tan \theta} \)

This simplification is a key step in leading us to verify the given trigonometric identity.

Tangent Identity

The tangent identity is another fundamental concept in trigonometry that involves the function tangent, typically expressed in terms of sine and cosine. This involves understanding how tangent relates to these functions.

Specifically, when using the tangent identity, we realize that:

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

This identity allows us to express tangent in terms of sine and cosine.

In the exercise, we make use of this identity by substituting \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) into our expression. This substitution is crucial in transforming the equation into terms that can be directly manipulated using basic algebraic techniques.

Quotient Identity

The quotient identity is an important piece of the trigonometric puzzle that connects the tangent function with sine and cosine. Understanding this identity helps us to simplify complex trigonometric expressions.

The quotient identity is expressed as:

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

This formula tells us that by knowing sine and cosine, we can easily find the tangent.

In solving the problem, this identity was used to rewrite \( \tan \theta \) in the expression \( \frac{1 - \tan \theta}{1 + \tan \theta} \), transforming it into \( \frac{1 - \frac{\sin \theta}{\cos \theta}}{1 + \frac{\sin \theta}{\cos \theta}} \). This step is foundational for further simplification which will ultimately match the identity we need to verify.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler forms. This is done using identities and algebraic manipulation.

In the exercise, the ultimate goal was to transform the expression \( \tan \left(\frac{\pi}{4} - \theta\right) \) into \( \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \).

We replace \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \) and perform algebraic simplification. By clearing the complex fraction, we combine terms over a common denominator:

• \( \frac{1 - \frac{\sin \theta}{\cos \theta}}{1 + \frac{\sin \theta}{\cos \theta}} \) simplifies to \( \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \)

Algebraic simplification is key here. By performing these steps, the complex expression becomes the original right-hand side of the identity, verifying the solution.