Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

Both sides match after simplification, verifying the identity.

Step 1: Understand the Identity

We need to show that the expression \( \sec \theta + \csc \theta - \cos \theta - \sin \theta \) is equivalent to the expression \( \sin \theta \tan \theta + \cos \theta \cot \theta \). Our goal is to transform one side of the equation into the other using trigonometric identities.

Step 2: Simplify the Left Side using Identities

The left-hand side is \( \sec \theta + \csc \theta - \cos \theta - \sin \theta \). Using the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \), rewrite it as: \( \frac{1}{\cos \theta} + \frac{1}{\sin \theta} - \cos \theta - \sin \theta \).

Step 3: Find a Common Denominator

Combine the terms with fractions by finding a common denominator. The common denominator for \( \frac{1}{\cos \theta} \) and \( \frac{1}{\sin \theta} \) is \( \sin \theta \cos \theta \). So we have: \( \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} - \cos \theta - \sin \theta \).

Step 4: Separate the Fraction and Simplify

Split the fraction: \( \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} = \frac{\sin \theta}{\sin \theta \cos \theta} + \frac{\cos \theta}{\sin \theta \cos \theta} = \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \). This simplifies to \( \tan \theta \sin \theta + \cot \theta \cos \theta \).

Step 5: Simplify Using Basic Trigonometric Identities

Now consider \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). The expression becomes: \( \sin \theta \frac{\sin \theta}{\cos \theta} + \cos \theta \frac{\cos \theta}{\sin \theta} \). This simplifies to \( \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\sin \theta} \).

Step 6: Verify Equality

Now, both sides have been simplified to the same form: \( \sin \theta \tan \theta + \cos \theta \cot \theta = \left( \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\sin \theta} \right) \). Therefore, the original identity is verified as both sides match.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the study of triangles and oscillatory phenomena. They relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These functions can be extended to secant (\( \sec \theta = \frac{1}{\cos \theta} \)), cosecant (\( \csc \theta = \frac{1}{\sin \theta} \)), and cotangent (\( \cot \theta = \frac{1}{\tan \theta} \)). Each of these functions plays a crucial role in analyzing angles and their relationships. They also assist in various mathematical transformations.

• Sine and cosine: Fundamental in defining other trig identities.
• Secant and cosecant: Reciprocals of cosine and sine, respectively.

Understanding these functions and their relationships helps in simplifying and verifying trigonometric identities.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics, allowing for more straightforward and elegant solutions. In the context of trigonometric identities, this often involves rewriting expressions using known identities. To simplify complex fractions or expressions:

• Identify common factors between terms.
• Use known trigonometric identities, like \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \), to rewrite terms appropriately.
• Find a common denominator to combine fractions.
• Group and reduce terms where possible.

Simplifying helps decrease the complexity of equations and makes verification easier. Mastering this concept is crucial when encountering larger expressions in trigonometry.

Verifying Identities

Verifying trigonometric identities involves demonstrating that two different expressions are, in fact, equivalent. This process requires manipulation and simplification using trusted trigonometric identities such as Pythagorean and reciprocal identities. To successfully verify an identity, follow these strategies:

• Start by simplifying one side of the equation, often the more complex side.
• Use identities and algebraic principles to transform expressions.
• Convert all terms to sine and cosine if necessary, since they're fundamental.
• Ensure both sides of the identity reach an identical form.

As you work through verifying identities, remember that the goal is to establish the equality of both sides precisely. This skill is invaluable for not only solving problems but also understanding deeper relationships in trigonometry.