Problem 6 - Precalculus with Calculus Previews

Question

Use the fundamental identities and the even-odd identities to simplify each expression. $$ 1+\tan ^{2}(-\theta) $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sec^2(\theta)\).

Step 1: Understand the Given Expression

The expression given is \(1 + \tan^2(-\theta)\). We need to use trigonometric identities to simplify it.

Step 2: Apply the Even-Odd Identity

Use the even-odd identity for tangent: \(\tan(-\theta) = -\tan(\theta)\). This means \(\tan^2(-\theta) = \tan^2(\theta)\) because \((-\tan(\theta))^2 = \tan^2(\theta)\). Therefore, the expression becomes \(1 + \tan^2(\theta)\).

Step 3: Apply the Pythagorean Identity

The Pythagorean identity states that \(1 + \tan^2(\theta) = \sec^2(\theta)\). Substitute this identity into the expression. Thus, the simplified form of \(1 + \tan^2(\theta)\) is \(\sec^2(\theta)\).

Key Concepts

Understanding Even-Odd Identities

Trigonometric functions have unique properties, which can simplify complex expressions. Even-odd identities are one of these useful tools. These identities describe how trigonometric functions behave with positive and negative inputs.
For tangent, the even-odd identity is

• \( \tan(-\theta) = -\tan(\theta) \)

This identity tells us that tangent is an odd function. The sign of tan changes when the angle is negative.
This is helpful because it reduces confusion over angle direction, making equations simpler to solve.
In our exercise, knowing \( \tan^2(-\theta) = \tan^2(\theta) \) lets us ignore the sign of \( \theta \) when squared. This streamlines our calculations by working with positive angles.

Exploring the Pythagorean Identity

The Pythagorean identity in trigonometry is akin to the Pythagorean theorem in geometry. It provides a fundamental relationship between trigonometric functions.
One of the key versions of this identity is:

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

This equation is derived from the core identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). By dividing every term by \( \cos^2(\theta) \), we get the Pythagorean identity for tangent and secant.
This relationship allows us to simplify expressions involving \( \tan^2(\theta) \), turning them into \( \sec^2(\theta) \).
This identity is invaluable when simplifying expressions, like in our exercise, resulting in cleaner solutions.

Understanding the Tangent Function

The tangent function is a crucial part of trigonometry, often abbreviated to \( \tan \). It helps relate angles in a right triangle to ratios of sides.
The tangent of angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Tangent can also be represented using sine and cosine:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

This connection shows how closely related these trigonometric functions are, and provides insight into their identities.
Understanding \( \tan(\theta) \) helps solve equations and prove identities, especially when using even-odd and Pythagorean identities. In our exercise, recognizing how \( \tan^2(-\theta) \) extracts to \( \tan^2(\theta) \) made the final transformation into \( \sec^2(\theta) \) straightforward.