Problem 27 - Algebra 2

Question

Simplify each trigonometric expression. $$ \cot \theta \tan \theta-\sec ^{2} \theta $$

Step-by-Step Solution

Answer

The simplified form of the given trigonometric expression \( \cot \theta \tan \theta - \sec^2 \theta \) is \( - \tan^2 \theta \).

Step 1: Simplify the Product of Cotangent and Tangent

By using the property \( \cot \theta \cdot \tan \theta = 1 \), you can simplify the expression to \(1 - \sec^2 \theta\).

Step 2: Apply Trigonometric Identity to Replace Secant

Recognize that \( \sec^2 \theta \) can be replaced with \( 1 + \tan^2 \theta \) by using the Pythagorean trigonometric identity \( \sec^2 \theta = 1 + \tan^2 \theta \). This gives us \( 1 - (1 + \tan^2 \theta) \).

Step 3: Simplify the Expression

By Simplifying the expression, we get \(1 - 1 - \tan^2 \theta = - \tan^2 \theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are like handy tools in a mathematician's toolbox. These are equations that are always true and involve trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant. Understanding these identities can make complex problems much simpler.

Here are some of the most commonly used identities:

• The Pythagorean Identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Reciprocal Identities, for instance, \( \csc \theta = \frac{1}{\sin \theta} \)
• Quotient Identities like \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Recognizing and applying these identities can simplify and solve expressions efficiently.
In the example, using \( \cot \theta \cdot \tan \theta = 1 \) allowed us to reduce the expression significantly.

Simplifying Expressions

Simplifying expressions in trigonometry means making them as simple as possible without changing their value. It often involves using the trigonometric identities to rewrite parts of the expression in a clearer or reduced form.

This process improves our ability to understand and manage the expressions more easily. Take the original expression \( \cot \theta \tan \theta - \sec^2 \theta \) as an example. By recognizing that \( \cot \theta \cdot \tan \theta = 1 \), we immediately simplified it to \(1 - \sec^2 \theta \).
From there, knowing the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) helped to transform and further simplify the expression to \(- \tan^2 \theta \).
This approach not only led to the solution but also demonstrated the efficiency of using identities correctly, making the process straightforward and less error-prone.

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry. It's called so because it stems from the Pythagorean theorem. One of its most widely used forms is: \( \sin^2 \theta + \cos^2 \theta = 1 \).
But, there are a couple of derived identities that are just as important:

• \(1 + \tan^2 \theta = \sec^2 \theta \)
• \(1 + \cot^2 \theta = \csc^2 \theta \)

In our exercise, the Pythagorean Identity \( \sec^2 \theta = 1 + \tan^2 \theta \) played a crucial role. By substituting \( \sec^2 \theta \) in our expression, we could directly simplify it to \(- \tan^2 \theta \). This illustrates the power of Pythagorean Identities in trigonometry - they not only simplify expressions but also reveal hidden connections between different trigonometric functions.