Problem 62 - Algebra and Trigonometry with Analytic Geometry

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 identity is verified: \((1-\sin^2 \theta)(1+\tan^2 \theta)=1\).

Step 1: Understand the Components of the Identity

The given identity to verify is \( (1 - \sin^2 \theta)(1 + \tan^2 \theta) = 1 \). Our goal is to transform the left-hand side (LHS) into the right-hand side (RHS) using trigonometric identities.

Step 2: Simplify the First Component

Start with the first part: \( 1 - \sin^2 \theta \). Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can rewrite \( 1 - \sin^2 \theta \) as \( \cos^2 \theta \).

Step 3: Simplify the Second Component

Now consider the second part: \( 1 + \tan^2 \theta \). Using the trigonometric identity \( \tan^2 \theta + 1 = \sec^2 \theta \), we can rewrite \( 1 + \tan^2 \theta \) as \( \sec^2 \theta \).

Step 4: Combine the Simplifications

Substitute the simplified expressions into the original identity: \( (\cos^2 \theta)(\sec^2 \theta) \). This simplifies to:\[ \cos^2 \theta \cdot \sec^2 \theta = \cos^2 \theta \cdot \left(\frac{1}{\cos^2 \theta}\right) \]

Step 5: Simplify the Expression

The expression \( \cos^2 \theta \cdot \left(\frac{1}{\cos^2 \theta}\right) \) simplifies to \( 1 \) because multiplying a function by its reciprocal yields 1. Thus, the left-hand side equals the right-hand side.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine functions. It states:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is essential because it helps simplify trigonometric expressions, as seen in exercises like the one given. In the exercise, we have the term \(1 - \sin^2 \theta\). By applying the Pythagorean Identity, we can simplify this expression to \(\cos^2 \theta\).
This simplification is beneficial because it converts the expression to a form that's easier to work with when proving or verifying identities. Learning to recognize and apply the Pythagorean Identity quickly is crucial for solving trigonometric problems efficiently. It's often used in conjunction with other identities to transform expressions. Remember, understanding how sine and cosine relate through this identity will simplify many otherwise complex trigonometric problems.

Trigonometric Simplification

Trigonometric simplification involves rewriting expressions to make them easier to understand or prove. In our problem, we simplified two expressions using known identities:

• \(1 - \sin^2 \theta\) was simplified to \(\cos^2 \theta\) using the Pythagorean Identity.
• \(1 + \tan^2 \theta\) was simplified to \(\sec^2 \theta\), utilizing another identity that states \(\tan^2 \theta + 1 = \sec^2 \theta\).

Simplifying trigonometric expressions generally involves identifying parts of the expression that can be replaced by simpler or equivalent forms through standard identities. These simplifications often translate more complex expressions or identities into much simpler forms, thus assisting in the process of solving equations or verifying identities.
It's helpful to become familiar with the main trigonometric identities, as they allow us to manipulate expressions more intuitively and efficiently. Simplification is a crucial skill in solving not only verification problems but also in practical applications like engineering and physics.

Verifying Trigonometric Identities

Verifying trigonometric identities means proving that both sides of an equation are indeed equal by transforming one side to match the other. In this task, the goal is to manipulate the left-hand side (LHS) so it becomes the same as the right-hand side (RHS), which is simply \(1\) in this case.To verify the identity:

• Start with transforming the LHS. We replaced \(1 - \sin^2 \theta\) with \(\cos^2 \theta\), using the Pythagorean identity.
• Then, \(1 + \tan^2 \theta\) was rewritten as \(\sec^2 \theta\).
• Substituting these into the LHS, the expression becomes \((\cos^2 \theta)(\sec^2 \theta) = 1\).
• Finally, recognize that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \), leading to simplification to \(\cos^2 \theta \cdot \frac{1}{\cos^2 \theta} = 1\).

This step-by-step approach is a systematic method for transforming and proving identities. Working through the transformation methodically ensures that each step logically follows from the previous one, thus confirming the identity accurately.
Developing the ability to verify trigonometric identities improves your understanding of how trigonometric functions interrelate and enhances your overall problem-solving skills in mathematics.