Problem 46 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ \frac{\tan \beta+\cot \beta}{\tan \beta}=\csc ^{2} \beta $$

Step-by-Step Solution

Answer

By substituting the trigonometric identities and simplifying the equation, we can verify that the left side of the given equation equals the right side, \( \csc^{2} \beta \).

Step 1: Understand basic trigonometric identities

Before starting, remember the following basic trigonometric identities: \[ \tan \beta = \frac{\sin \beta}{\cos \beta}\] \[ \cot \beta = \frac{\cos \beta}{\sin \beta}\] \[ \csc ^{2} \beta = 1 + \cot^{2} \beta = \frac{1}{\sin^2 \beta}\]

Step 2: Substitute \(\tan \beta\) and \(\cot \beta\) using Basic Identities

Replace \(\tan \beta\) and \(\cot \beta\) in the equation with their equivalents in terms of sine and cosine: \[ \frac{\sin \beta / \cos \beta+ \cos \beta / \sin \beta}{\sin \beta / \cos \beta}\]

Step 3: Simplify

Simplify the fraction to bring it into a more understandable form. Multiply all terms by \( \sin\beta \cos \beta\) to clear the denominators: \[ \frac{(\sin^2 \beta + \cos^2 \beta )}{\sin \beta }\]

Step 4: Use Pythagorean Trigonometric Identity

We know that \(\sin^2 \beta + \cos^2 \beta‎ \) equals 1. Substitute this into the equation: \[ \frac{1}{\sin \beta }\]

Step 5: Transform into \(\csc ^{2} \beta\)

Notice that \( \frac{1}{\sin \beta }\) is equal to \( \csc \beta \). To square this, we simply: \[ \csc^{2} \beta\]

Key Concepts

Trigonometric Transformations

Trigonometric transformations involve the process of changing the form of a trigonometric expression without altering its value. These transformations rely on the understanding of basic trigonometric functions and identities.

Let's consider a scenario where you have an equation involving trigonometric functions and you're required to rewrite it in a different form. The goal is to manipulate the expression so that it becomes more workable or to match a given form. This is useful in solving trigonometric equations, simplifying expressions, and even in integration and differentiation.

In our example, we started with a complex fraction involving \(\tan\beta\) and \(\cot\beta\). Transforming these using their fundamental relationships with \(\sin\beta\) and \(\cos\beta\) helped to move towards simplification. By applying these transformations correctly, we are utilizing a foundational tool in trigonometry to proceed towards our end goal of matching the given right side of the equation.

Pythagorean Trigonometric Identity

The Pythagorean trigonometric identity is a cornerstone of trigonometry, representing the inherent relationship between the sine and cosine functions for any angle. It is expressed as:
\[ \sin^2\beta + \cos^2\beta = 1 \]
Understanding this identity is essential when trying to simplify trigonometric expressions, because it links the squares of \(\sin\beta\) and \(\cos\beta\), allowing for substitutions that can often simplify an otherwise complex expression.

In the provided exercise, the Pythagorean identity was used at a crucial step to simplify \(\sin^2\beta + \cos^2\beta\) into 1. This greatly reduced the complexity of the fraction and pushed us towards the final simplified form. Remember, the Pythagorean identity is just one of many identities in trigonometry, but it's among the most frequently used due to its fundamental nature.

Trigonometric Simplification

Trigonometric simplification is the process of breaking down complex trigonometric expressions into simpler forms. This not only makes the expressions easier to understand but also facilitates their manipulation in equations and calculus.

The key to effective simplification lies in familiarity with trigonometric identities and using them strategically. For example, transforming \(\tan\beta\) and \(\cot\beta\) into their respective \(\sin\beta\) and \(\cos\beta\) components and finding common denominators can help to merge terms. In the exercise, multiplication by the least common denominator, \(\sin\beta \cos\beta\), cleared the fractions allowing us to apply the Pythagorean identity.

Efficient simplification leads us to the finish line, where the complex starting expression evolved into the much neater \(\csc^2\beta\), showcasing trigonometry's power to reveal simplicity within complexity.