Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

Verified step by step guidance

Step 1: Start by simplifying the left-hand side of the equation. Combine the fractions \( \frac{1 + \sin \theta}{1 - \sin \theta} - \frac{1 - \sin \theta}{1 + \sin \theta} \) by finding a common denominator, which is \((1 - \sin \theta)(1 + \sin \theta)\).

Step 2: Rewrite each fraction with the common denominator: \( \frac{(1 + \sin \theta)^2 - (1 - \sin \theta)^2}{(1 - \sin \theta)(1 + \sin \theta)} \).

Step 3: Expand the numerators: \((1 + \sin \theta)^2 = 1 + 2\sin \theta + \sin^2 \theta\) and \((1 - \sin \theta)^2 = 1 - 2\sin \theta + \sin^2 \theta\).

Step 4: Subtract the expanded forms: \(1 + 2\sin \theta + \sin^2 \theta - (1 - 2\sin \theta + \sin^2 \theta)\), which simplifies to \(4\sin \theta\).

Step 5: The denominator \((1 - \sin \theta)(1 + \sin \theta)\) simplifies to \(1 - \sin^2 \theta = \cos^2 \theta\). Thus, the left-hand side becomes \(\frac{4\sin \theta}{\cos^2 \theta}\), which simplifies to \(4 \tan \theta \sec \theta\), verifying the identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations as identities.

Simplifying Expressions

Simplifying trigonometric expressions involves manipulating the equation using algebraic techniques and trigonometric identities to make it easier to analyze or prove. This may include combining fractions, factoring, or using identities to rewrite terms. Mastery of simplification techniques is essential for verifying the validity of trigonometric equations.

Tangent and Secant Functions

The tangent (tan) and secant (sec) functions are fundamental trigonometric functions defined as tan θ = sin θ / cos θ and sec θ = 1 / cos θ, respectively. These functions are often used in identities and equations involving angles. Understanding their relationships and how to manipulate them is key to solving and verifying trigonometric identities.

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