Problem 62 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\left(1-\sin ^{2} \theta\right)\left(1+\tan ^{2} \theta\right)=1$$

Step-by-Step Solution

Answer

The given identity is verified to be true.

Step 1: Expanding Terms

The expression given is \((1 - \sin^2 \theta)(1 + \tan^2 \theta)\). Start by expanding the left-hand side. We know from the Pythagorean identity that \(1 - \sin^2 \theta = \cos^2 \theta\). So, the expression becomes \(\cos^2 \theta (1 + \tan^2 \theta)\).

Step 2: Using the Pythagorean Identity for Tangent

Recall the identity \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\). Substituting this in, we get \(\cos^2 \theta (1 + \frac{\sin^2 \theta}{\cos^2 \theta})\).

Step 3: Simplifying the Expression

Simplify the expression by distributing \(\cos^2 \theta\). This gives us \(\cos^2 \theta + \sin^2 \theta\).

Step 4: Using the Pythagorean Identity Again

Use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this in to get the right-hand side as \(1\).

Step 5: Conclusion: Verify Identity

Since we successfully transformed the left side \((1 - \sin^2 \theta)(1 + \tan^2 \theta)\) into \(1\), we have verified the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is fundamental in trigonometry and is particularly helpful when verifying trigonometric identities. The most common form of this identity is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity stems from the Pythagorean theorem in geometry and reflects the constant relationship between the sine and cosine of an angle. In many trigonometric problems, like the one in the original exercise, this identity helps transform or simplify expressions. For instance, from \( \sin^2 \theta + \cos^2 \theta = 1 \), we can derive other essential identities:

• \( 1 - \sin^2 \theta = \cos^2 \theta \)
• \( 1 - \cos^2 \theta = \sin^2 \theta \)

These transformations allow us to substitute and rearrange terms easily, playing a crucial role in manipulating the given expressions to verify identities.

Trigonometric Simplification

Trigonometric simplification involves expressing trigonometric functions in their simplest form. When dealing with a complex expression, like the one in our exercise, understanding how to simplify effectively is key to tackling and verifying identities. Here's the general approach:1. **Identify Applicable Identities:** Spot identities such as Pythagorean identity or quotient identity (like \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)).2. **Substitute and Reduce:** Use these identities to substitute equivalent terms in the expression to make it less complex.3. **Combine Like Terms:** After substituting, combine like terms to further simplify.In our specific exercise:

• First, we used the Pythagorean identity to convert \(1 - \sin^2 \theta\) into \(\cos^2 \theta\).
• Then, we applied the identity \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\), allowing us to transform \( \tan^2 \theta \) terms.
• Finally, combined the terms \( \cos^2 \theta + \sin^2 \theta \) utilizing the fundamental Pythagorean identity to conclude the simplification.

This approach of simplifying by substitution and identity application is invaluable for trigonometric expressions.

Angle Transformation

Angle transformation refers to altering an angle's representation using various trigonometric identities.This is not only essential for simplifying expressions but also offers deeper insights into how these functions interrelate. Examples of angle transformations include identities like:

• \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)
By employing these transformations, we have the ability to rewrite expressions. This is particularly useful when certain terms need to be eliminated or simplified, as seen in the exercise.In our example, the transformation from \( \tan^2 \theta \) to \( \frac{\sin^2 \theta}{\cos^2 \theta} \) was pivotal in simplifying and manipulating the expression to verify the identity. By understanding these transformations, you enrich your ability to handle and manipulate trigonometric identities more flexibly and confidently.