Problem 97 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\cot \theta-\tan \theta=\frac{2 \cos ^{2} \theta-1}{\sin \theta \cos \theta}$$

Step-by-Step Solution

Answer

Yes, the given equation is an identity.

Step 1: Rewrite Cotangent and Tangent

Start by using the definitions of cotangent and tangent. Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This gives:\[ \cot \theta - \tan \theta = \frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta} \]

Step 2: Find a Common Denominator

Combine the fractions by finding a common denominator, which is \( \sin \theta \cos \theta \). Rewrite each term to use this common denominator:\[ \frac{\cos^2 \theta}{\sin \theta \cos \theta} - \frac{\sin^2 \theta}{\sin \theta \cos \theta} \]Combining these fractions results in:\[ \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} \]

Step 3: Apply the Cosine Double-Angle Identity

Recognize that the expression for \( \cos^2 \theta - \sin^2 \theta \) matches the identity for \( \cos 2\theta \):\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]Substitute this identity into the equation:\[ \frac{\cos 2\theta}{\sin \theta \cos \theta} \]

Step 4: Use the Cosine Double-Angle Identity Alternate Form

Recall another form of the double-angle identity: \( \cos 2\theta = 2\cos^2 \theta - 1 \).Since the problem also presents \( \frac{2\cos^2 \theta - 1}{\sin \theta \cos \theta} \) on the right side, recognize these as equivalent via the identity substitution.

Step 5: Confirm the Identity

Both sides of the equation now simplify to the same expression, confirming the identity:\[ \frac{\cos 2\theta}{\sin \theta \cos \theta} = \frac{2\cos^2 \theta - 1}{\sin \theta \cos \theta} \]Thus, the original statement \( \cot \theta - \tan \theta = \frac{2\cos^2 \theta - 1}{\sin \theta \cos \theta} \) is indeed an identity.

Key Concepts

Cotangent and Tangent Definitions

Understanding cotangent and tangent is crucial for solving trigonometric equations and identities. To start, the cotangent of an angle \( \theta \) is defined as the ratio of the cosine to the sine of that angle. In mathematical terms, this means \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). The tangent of an angle \( \theta \) is the inverse of this ratio, or more simply, the sine to cosine ratio, expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
Let's break this down further:

• \( \cot \theta \) is used to understand relationships where the cosine angle dominates over sine.
• \( \tan \theta \) provides insight into angles where the sine has a greater influence.

By comprehending these definitions, you'll be able to manipulate and understand equations involving these trigonometric functions more effectively. These ratios allow you to reframe and solve equations more efficiently, as seen in the exercise with \( \cot \theta - \tan \theta \).

Double-Angle Identity

The double-angle identities are powerful tools in trigonometry, especially when simplifying expressions or verifying identities. A double-angle identity essentially provides a formula to express trigonometric functions of double angles (like \( 2\theta \)) in terms of the functions of \( \theta \).
In our exercise, we utilize the cosine double-angle identity. This states that \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). This identity was key to transforming the original expression into a simpler form.
Another useful form of the identity, often seen in different types of problems, is \( \cos 2\theta = 2\cos^2 \theta - 1 \). Both versions help in linking complex expressions to a more manageable form, making calculations and proofs easier.

• The identity \( \cos^2 \theta - \sin^2 \theta \) simplifies expressions and often relates directly to other identities.
• Recognizing these formulas can help spot patterns in seemingly complicated equations.

Mastery of these identities enriches problem-solving skills and deepens understanding of trigonometric relationships.

Trigonometric Equations

Trigonometric equations involve expressions equalized through trigonometric functions like sine, cosine, tangent, and so on. These equations are not only solved for specific values but often require confirming or verifying identities, as with the exercise given.
Key strategies for solving such equations include:

• Identify known identities or formulas that might apply, like Pythagorean or double-angle identities.
• Manipulate the equation into simpler fractions or expressions, as seen when achieving common denominators.
• Re-cast or substitute parts of the equation using identities, allowing simplification or solving for specific variables.

Tackling trigonometric equations often means resolving each part into its base sine and cosine components. Such simplification can clarify the path to the solution, making complex equations more approachable.
The exercise you encountered demonstrates eloquently how combining multiple identities and strategic manipulations confirm that an equation holds as true, or is indeed an identity.