Problem 57 - Algebra 2

Question

Verify each identity. $$ \frac{\cot \theta \sin \theta}{\sec \theta}+\frac{\tan \theta \cos \theta}{\csc \theta}=1 $$

Step-by-Step Solution

Answer

The given expression simplifies to \(2\), not \(1\). The original equation appears to be incorrect.

Step 1: Substitute Trigonometric Identities

Rewrite the expression in terms of simpler trigonometric functions using the equivalents \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). When that's done, the expression becomes \(\frac{\cos \theta}{\cos \theta} + \frac{\sin \theta}{\sin \theta}\).

Step 2: Simplify the Expression

Simplify the terms. \(\frac{\cos \theta}{\cos \theta}\) simplifies to 1 and \(\frac{\sin \theta}{\sin \theta}\) simplifies to 1.

Step 3: Combine the Terms

The expression has now been simplified to \(1+1\), which is easy to evaluate.

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine, cosine, tangent, secant, cosecant, and cotangent form the backbone of trigonometry. These functions relate the angles of a triangle to the lengths of its sides. They are crucial in various fields like physics, engineering, and even in everyday problem-solving. Here's a brief introduction:

• Sine (\( \sin \theta \)): This function gives the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (\( \cos \theta \)): This describes the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): This is the ratio of the sine to the cosine, essentially opposite side over adjacent side.
• Cosecant, Secant, and Cotangent: These are the reciprocals of sine, cosine, and tangent respectively.
  Cosecant (\( \csc \theta \)) is \( \frac{1}{\sin \theta} \), Secant (\( \sec \theta \)) is \( \frac{1}{\cos \theta} \), and Cotangent (\( \cot \theta \)) is \( \frac{1}{\tan \theta} \) or \( \frac{\cos \theta}{\sin \theta} \).

Understanding these relationships helps you easily transform one function into another, as seen in the conversion within the original exercise.

Simplifying Expressions

Simplifying expressions makes complex mathematical statements easier to understand and solve. It involves reducing expressions to their simplest form by applying arithmetic operations and mathematical properties. In trigonometry, simplification often means using trigonometric identities to replace functions with more manageable equivalents. For example, replacing \(\sec \theta\) with \(\frac{1}{\cos \theta}\) or \(\csc \theta\) with \(\frac{1}{\sin \theta}\).

• The original problem replaces complex trigonometric expressions with basic identities.
• Through substitution, \(\frac{\cot \theta \sin \theta}{\sec \theta}\) reduces down to \(\frac{\cos \theta}{\cos \theta}\), simplifying directly to 1.
• Similarly, refining \(\frac{\tan \theta \cos \theta}{\csc \theta}\) reduces to \(\frac{\sin \theta}{\sin \theta}\), also simplifying to 1.
• These simplifications render the expression as 1 + 1, a clear and simple form.

To master simplification, practice with various identities to find the easiest ways to reconfigure expressions.

Mathematical Proofs

Mathematical proofs are rigorous arguments that justify why a statement is true. In trigonometry, verifying identities often involves proving that two expressions are equivalent by transforming them using identities and simplifications. Proofs require logic and a deep understanding of the properties involved.
A proof aims to show, step-by-step, how an expression which may appear complex equates to a simpler or more recognizable form. In the context of our task:

• We began with the expression \( \frac{\cot \theta \sin \theta}{\sec \theta} + \frac{\tan \theta \cos \theta}{\csc \theta} \).
• By rewriting using trigonometric identities, each part was broken down to simpler terms.
• After simplification, the entire expression was shown to equal 2, thus proving the identity.

Building proofs reinforces your grasp on how different parts of mathematics interconnect. Effective proofs elucidate the path from initial expressions to conclusions, and continuous practice hones this critical skill.