Problem 146 - Calculus Volume 1

Question

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. $$\frac{1+\tan ^{2} \alpha}{1+\cot ^{2} \alpha}$$

Step-by-Step Solution

Answer

The simplified expression is \( \tan^2 \alpha \).

Step 1: Express terms in sine and cosine

We start by expressing tangent and cotangent in terms of sine and cosine. Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \). Substitute these into the expression: \[ \frac{1+\left(\frac{\sin \alpha}{\cos \alpha}\right)^2}{1+\left(\frac{\cos \alpha}{\sin \alpha}\right)^2} \]

Step 2: Simplify each fraction

Simplify the terms inside the square parentheses: \[ \frac{1+\frac{\sin^2 \alpha}{\cos^2 \alpha}}{1+\frac{\cos^2 \alpha}{\sin^2 \alpha}} = \frac{\frac{\cos^2 \alpha + \sin^2 \alpha}{\cos^2 \alpha}}{\frac{\sin^2 \alpha + \cos^2 \alpha}{\sin^2 \alpha}} \]

Step 3: Apply the Pythagorean identity

Employ the Pythagorean identity, \( \sin^2 \alpha + \cos^2 \alpha = 1 \), in both the numerator and the denominator: \[ \frac{\frac{1}{\cos^2 \alpha}}{\frac{1}{\sin^2 \alpha}} \]

Step 4: Simplify the complex fraction

To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator: \[ \frac{1}{\cos^2 \alpha} \times \frac{\sin^2 \alpha}{1} = \frac{\sin^2 \alpha}{\cos^2 \alpha} \]

Step 5: Express the final answer

Recognize that \( \frac{\sin^2 \alpha}{\cos^2 \alpha} \) simplifies to \( \tan^2 \alpha \). Thus, the simplified form of the original expression is \( \tan^2 \alpha \).

Key Concepts

Simplifying Expressions

Simplifying expressions is an important technique in mathematics, especially in trigonometry. It allows us to express complex expressions in simpler forms. The process often involves replacing trigonometric functions or identities with simpler terms, like using basic operations instead of complex fractions. In trigonometry, we often express functions in terms of sine and cosine to make them easier to handle.
When you are given an expression that involves trigonometric functions like tangent and cotangent, the first step to simplify is to rewrite these functions in terms of the more basic sine and cosine. This helps in manipulating the expression, as sine and cosine are the fundamental trigonometric functions.
After rewriting the expression, the next step might involve simplifying fractions or solving complex fractions. Always look for opportunities to cancel terms or apply known identities, which can significantly simplify your results. Finally, check if the resulting expression can be further simplified using algebraic techniques or additional trigonometric identities. This step ensures that the expression is in its simplest form.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that play a crucial role in simplifying expressions. The sine function, denoted as \( \sin \alpha \), is defined as the ratio of the opposite side to the hypotenuse in a right triangle. Similarly, cosine, \( \cos \alpha \), is the ratio of the adjacent side to the hypotenuse. These two functions are at the core of trigonometry and provide the basis for many other trigonometric functions.
When working with expressions involving other trigonometric functions, converting them to sine and cosine can greatly simplify calculations. For example, the tangent function \( \tan \alpha \) can be rewritten as \( \frac{\sin \alpha}{\cos \alpha} \), and the cotangent function \( \cot \alpha \) as \( \frac{\cos \alpha}{\sin \alpha} \). This conversion is often a critical step in simplifying expressions, as seen in our original exercise.
Understanding sine and cosine's relationships with other functions allows for easier manipulation and simplification, often revealing hidden relationships or identities that can be useful in further steps.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that states that for any angle \( \alpha \), \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This identity is derived from the Pythagorean theorem and is a foundational concept in trigonometry, frequently used in simplifying expressions and proving other identities.
In the context of simplifying expressions, like in the original exercise, we use this identity to recognize expressions that can be replaced with 1. For example, in both the numerator and the denominator of our expression, \( \sin^2 \alpha + \cos^2 \alpha \) simplifies immediately to 1. This simplification is crucial in transforming complex expressions into much easier forms.
Using the Pythagorean identity can also help verify the correctness of your simplifications. It's a powerful tool to simplify complex trigonometric expressions to more manageable forms or even to derive other identities. This identity's simplicity makes it invaluable in various trigonometric applications, reinforcing the understanding of trigonometric functions' interdependence.