Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\cot \theta+\tan \theta=\csc \theta \sec \theta$$

Step-by-Step Solution

Answer

The identity is verified: \( \cot \theta + \tan \theta = \csc \theta \sec \theta \).

Step 1: Convert to Sine and Cosine

Express cotangent and tangent in terms of sine and cosine. We have \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute these into the left-hand side of the equation: \[\cot \theta + \tan \theta = \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta}.\]

Step 2: Find a Common Denominator

To add these fractions, we need a common denominator, which is \( \sin \theta \cdot \cos \theta \). Rewrite each term with this common denominator:\[\frac{\cos \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta \cdot \cos \theta}, \quad \frac{\sin \theta}{\cos \theta} = \frac{\sin^2 \theta}{\sin \theta \cdot \cos \theta}.\]

Step 3: Combine Fractions

Add the fractions obtained from Step 2 with the common denominator:\[\frac{\cos^2 \theta}{\sin \theta \cdot \cos \theta} + \frac{\sin^2 \theta}{\sin \theta \cdot \cos \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cdot \cos \theta}.\]

Step 4: Use Pythagorean Identity

Recall the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \). Apply this identity to simplify the numerator:\[\frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cdot \cos \theta} = \frac{1}{\sin \theta \cdot \cos \theta}.\]

Step 5: Express in Terms of Cosecant and Secant

Recognize that \( \frac{1}{\sin \theta} = \csc \theta \) and \( \frac{1}{\cos \theta} = \sec \theta \). Thus,\[\frac{1}{\sin \theta \cdot \cos \theta} = \csc \theta \cdot \sec \theta.\]

Step 6: Verify the Identity

We have transformed the left-hand side \( \cot \theta + \tan \theta \) into \( \csc \theta \cdot \sec \theta \), which matches the right-hand side of the identity. Therefore, the given identity is verified.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry and can be expressed as: \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity stems from the Pythagorean theorem related to right triangles on the unit circle. Here is why the Pythagorean Identity is so useful:

• Firstly, it helps in simplifying expressions where both sine and cosine are present.
• Secondly, it can be used to verify and prove other trigonometric identities.
• Lastly, in problems where either sine or cosine values are given, it makes finding the other value straightforward.

In our example, when we combined the terms \( \cos^2 \theta + \sin^2 \theta \) in the exercise, applying the Pythagorean Identity allowed us to simplify the expression to 1. This simplification is key to transforming complex trigonometric expressions into more manageable forms, leading to solutions that can be easily verified.

Sine and Cosine Conversion

Trigonometric functions like tangent and cotangent can often be expressed in terms of sine and cosine. This conversion is especially useful when trying to simplify or manipulate trigonometric equations. Let's break it down:

• The cotangent of an angle, \( \theta \), is defined as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• Similarly, the tangent of an angle is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

By expressing tangent and cotangent this way, we can leverage the properties of sine and cosine to work with the functions more directly. In the exercise, converting \( \cot \theta \) and \( \tan \theta \) into their respective sine and cosine forms was an essential step in allowing us to later add these fractions together successfully.These conversions create a common ground for simplifying, solving, or even proving identities in trigonometry.

Common Denominator in Fractions

When adding or subtracting fractions, finding a common denominator is a crucial part of the process. In arithmetic, just as in trigonometry, this makes it possible to combine terms or simplify expressions. Consider this process:

• To add \( \frac{\cos \theta}{\sin \theta} \) and \( \frac{\sin \theta}{\cos \theta} \), a common denominator must be established.
• The least common denominator here is \( \sin \theta \cdot \cos \theta \), which enables straightforward addition of the fractions.

After rewriting each fraction with the common denominator, we can seamlessly combine them into a single fraction. In the exercise, adopting a common denominator paved the way for simplifying the original expression into a unified form. This technique is essential not only for proving identities but also for solving equations and simplifying expressions across various mathematical disciplines.