Problem 26 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \frac{\cos \theta}{\sec \theta}+\frac{\sin \theta}{\csc \theta}=1 $$

Step-by-Step Solution

Answer

The equation is an identity because \( \cos^2 \theta + \sin^2 \theta = 1 \).

Step 1: Simplify Sine and Cosine Functions

Write the equation using the basic trigonometric identities. Recall that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Substitute these into the equation:\[ \frac{\cos \theta}{\sec \theta} + \frac{\sin \theta}{\csc \theta} = \frac{\cos \theta}{\frac{1}{\cos \theta}} + \frac{\sin \theta}{\frac{1}{\sin \theta}} \]

Step 2: Simplify Fractions

Simplify the fractions \( \frac{\cos \theta}{\frac{1}{\cos \theta}} \) and \( \frac{\sin \theta}{\frac{1}{\sin \theta}} \) by multiplying the numerator by the reciprocal of the denominator:\[ \cos \theta \cdot \cos \theta + \sin \theta \cdot \sin \theta \]

Step 3: Use the Pythagorean Identity

Use the Pythagorean Identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to simplify the expression:\[ \cos^2 \theta + \sin^2 \theta = 1 \]

Step 4: Conclude the Proof

Since \( \cos^2 \theta + \sin^2 \theta = 1 \) holds true for all \( \theta \), the original equation is proven to be an identity:\[ \frac{\cos \theta}{\sec \theta} + \frac{\sin \theta}{\csc \theta} = 1 \]

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \): \[ \cos^2 \theta + \sin^2 \theta = 1 \] This identity is derived from the Pythagorean Theorem and links the sine and cosine functions. Its simplicity belies its power, as it is used to prove various other identities and solve trigonometric equations. Whenever you see expressions like \( \cos^2 \theta + \sin^2 \theta \) in an equation, you can replace them with 1. Understanding this identity will help you simplify complex trigonometric expressions with ease. It's very common in math problems where you need to prove something to be always true or identify a pattern. It's like a magic formula that translates complicated trigonometric officials into something readily understandable.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are essential for transforming and simplifying trigonometric expressions. They include:

• Secant: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cosecant: \( \csc \theta = \frac{1}{\sin \theta} \)
• Cotangent: \( \cot \theta = \frac{1}{\tan \theta} \)

In the given exercise, understanding these reciprocal identities allows us to simplify expressions like \( \frac{\cos \theta}{\sec \theta} \). This simplification is key to breaking down complex problems into manageable parts. By substituting the reciprocal identities, you transform a trigonometric fraction into a simpler form. This kind of manipulation often leads you one step closer to identifying an equation or proving an identity.

Simplification in Trigonometry

Simplification in trigonometry focuses on reducing expressions to their simplest form. This exercise is a perfect example of how simplification can be systematically achieved. Initially, we face fractions like \( \frac{\cos \theta}{\sec \theta} \). Simplifying these involves multiplying by the reciprocal of the denominator. Through simplification, what seemed complicated initially becomes clear and manageable. The idea is to break the problem down using known identities and straightforward algebraic manipulations:

• Identify parts of the expression that match trigonometric identities.
• Simplify fractions by recognizing equivalent expressions.
• Apply known identities like the Pythagorean Identity to make a concise conclusion.

Such methods not only prove identities but also make equations easier to work with. By habitual practice, you will be able to spot opportunities for simplification quickly, enhancing your problem-solving skills.