Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\cot \theta-\tan \theta}{\sin \theta+\cos \theta}=\csc \theta-\sec \theta $$

Step-by-Step Solution

Answer

The identity is verified as both sides are equal.

Step 1: Rewrite trigonometric functions using basic identities

First, recall the trigonometric identities: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute these into the left side of the identity:\[\frac{\cot \theta - \tan \theta}{\sin \theta + \cos \theta} = \frac{\frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta}}{\sin \theta + \cos \theta}.\]

Step 2: Find a common denominator for numerator expressions

Combine the terms in the numerator by finding a common denominator, which would be \(\sin \theta \cos \theta\):\[\frac{\frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta}}{\sin \theta + \cos \theta}.\]

Step 3: Simplify the complex fraction

The expression becomes a complex fraction. Simplify it by multiplying the numerator and denominator by \(\sin \theta \cos \theta\):\[\frac{\cos^2 \theta - \sin^2 \theta}{(\sin \theta + \cos \theta)(\sin \theta \cos \theta)}.\]

Step 4: Simplify the numerator using identities

Use the identity \(\cos^2 \theta - \sin^2 \theta = (\cos \theta + \sin \theta)(\cos \theta - \sin \theta)\), allowing further simplification:\[\frac{(\cos \theta + \sin \theta)(\cos \theta - \sin \theta)}{(\sin \theta + \cos \theta)(\sin \theta \cos \theta)}.\]

Step 5: Cancel common terms and simplify

Cancel the term \(\cos \theta + \sin \theta\) from both numerator and denominator:\[\frac{\cos \theta - \sin \theta}{\sin \theta \cos \theta}.\] This simplifies further to:\[\frac{1}{\sin \theta} - \frac{1}{\cos \theta} = \csc \theta - \sec \theta.\]

Step 6: Final verification

Both sides of the original equation now match:\[\csc \theta - \sec \theta = \csc \theta - \sec \theta.\]This confirms that the identity is verified.

Key Concepts

Cotangent and Tangent Identities

Understanding cotangent and tangent identities is fundamental in trigonometry, especially when solving problems that require trigonometric simplification. The cotangent (\(\cot \theta\)) and tangent (\(\tan \theta\)) of an angle are reciprocals of one another, expressed using sine and cosine functions:

• The cotangent is \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
• The tangent is \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

These identities are useful because they let us represent tangent and cotangent in terms of sine and cosine, simplifying expressions by reducing them to these basic trigonometric functions.
In problems involving the combination of cotangent and tangent, we often use their reciprocal relationships to find common denominators or to tie different parts of an equation together. Being comfortable with these identities allows you to transition smoothly between different trigonometric forms, which is crucial in both simplifying and verifying identities.

Simplifying Complex Fractions

Simplifying complex fractions is an essential skill when dealing with trigonometric identities and expressions. A complex fraction is one where the numerator or the denominator (or both) are themselves fractions.
To simplify, you first identify a common denominator for the fractions involved. In this exercise, \(\sin \theta \cos \theta\) was used as a common denominator to combine \(\cot \theta\) and \(\tan \theta\):

• Write the expression with a single denominator for the numerator. This involves subtracting \(\frac{\sin \theta}{\cos \theta}\) from \(\frac{\cos \theta}{\sin \theta}\)
• The result becomes \(\frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta}\).
• Simplify further by multiplying both the top and bottom of the complex fraction by the least common denominator.

This step reduces the complexity of the fraction and makes it easier to work with.

Trigonometric Simplification

Trigonometric simplification involves using known identities to transform complex trigonometric expressions into simpler forms. This is particularly useful in verifying identities, as it helps to reveal connections between different sides of an equation.

• Apply the Pythagorean identities like \(\cos^2 \theta - \sin^2 \theta\) as \((\cos \theta + \sin \theta)(\cos \theta - \sin \theta)\)
• Use factorization to break down expressions into simpler multiplicative components.
• Cancel matching terms in the numerator and denominator.

In this exercise, breaking down \(\cos^2 \theta - \sin^2 \theta\) and canceling like terms help realize that one side of the equation can be rewritten to match the other. This makes verifying the identity much more straightforward.

Verifying Trigonometric Identities

Verifying trigonometric identities involves confirming that two sides of an equation are equivalent through algebraic and trigonometric manipulations. It requires a deep understanding of identities and simplification techniques to establish the equality confidently.

• Start with one side of the equation and manipulate it until it resembles the other side.
• Use known identities and simplifications such as reciprocal relationships and Pythagorean identities to assist in the transformation.
• Multiply or divide strategically to simplify complex expressions without changing their values.

In our original problem, simplifying each side independently and confirming they equal \(\csc \theta - \sec \theta\) verifies the given identity. Successful verification strengthens your understanding of how trigonometric concepts interrelate.