Problem 5 - Precalculus with Calculus Previews

Question

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \tan ^{2} t-\sec ^{2} t $$

Step-by-Step Solution

Answer

The simplified expression is \( -1 \).

Step 1: Recall the Pythagorean identity

One of the fundamental trigonometric identities is \( 1 + \tan^2 \theta = \sec^2 \theta \). This identity will be useful for simplifying the expression.

Step 2: Substitute the identity into the expression

The given expression is \( \tan^2 t - \sec^2 t \). According to the identity \( 1 + \tan^2 t = \sec^2 t \), we can rewrite \( \sec^2 t \) as \( 1 + \tan^2 t \).

Step 3: Simplify the expression

Substitute \( \sec^2 t \) with \( 1 + \tan^2 t \) in the expression: \[ \tan^2 t - \sec^2 t = \tan^2 t - (1 + \tan^2 t) \].Distribute the negative through the parentheses: \[ \tan^2 t - 1 - \tan^2 t \].This simplifies to \( -1 \) because \( \tan^2 t - \tan^2 t = 0 \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a cornerstone in the world of trigonometry. It connects the trigonometric functions tangent and secant through the equation:

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

This identity is instrumental in simplifying trigonometric expressions. It allows one function to be expressed in terms of another, creating simpler forms for complex expressions.
The identity essentially arises from the Pythagorean theorem when you consider a right-angled triangle inscribed in a unit circle. In practical terms, whenever you encounter a trigonometric expression involving \( \tan^2 \theta \) and \( \sec^2 \theta \), checking if this identity can be used might drastically simplify your work.
Always look for ways to substitute the terms such that these relationships can help reduce the expression to its simplest form.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often turns out to be straightforward once you apply the right identities or substitutions. The primary goal is to reduce expressions to a form that is easier to interpret.
Start by identifying known identities like the Pythagorean identity, angle sum formulas, or double angle identities. For a given expression, check whether any trigonometric identity can replace one complex part of the expression.
For instance, in our original expression \( \tan^2 t - \sec^2 t \), we can notice a direct application of the Pythagorean identity. Replacing \( \sec^2 t \) with \( 1 + \tan^2 t \) gives us a simpler result after performing arithmetic operations, transforming the complex expression into a simple constant, \( -1 \).
Essentially, simplifying relies heavily on familiarity with trigonometric formulas and practice in substituting terms effectively.

Even-Odd Identities

Even-odd identities in trigonometry help distinguish how certain functions behave under negation inputs. This is crucial for understanding angles in different quadrants and simplifying expressions involving both positive and negative angles.
Here's a quick guide to remember which functions are odd, meaning they satisfy \( f(-\theta) = -f(\theta) \), and which are even, meaning they satisfy \( f(-\theta) = f(\theta) \):

• Even Functions: \( \cos \theta, \sec \theta \)
• Odd Functions: \( \sin \theta, \csc \theta, \tan \theta, \cot \theta \)

Recognizing this behavior can be particularly useful when dealing with expressions that include negative angles or transformations.
Although not directly required in simplifying our original exercise, it's a valuable tool in the trigonometric toolbox. By understanding the symmetry of functions around the origin or the y-axis, you can simplify expressions across different domains effectively.