Verify that each equation is an identity.
[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ

Verified step by step guidance

1. Step 1: Start by simplifying the left-hand side (LHS) of the equation. Expand \((\sec \theta - \tan \theta)^2\) using the identity \((a - b)^2 = a^2 - 2ab + b^2\).
2. Step 2: Use the Pythagorean identities \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(\csc^2 \theta = 1 + \cot^2 \theta\) to simplify the expression further.
3. Step 3: Simplify the denominator \(\sec \theta \csc \theta - \tan \theta \csc \theta\) by expressing \(\sec \theta\) and \(\csc \theta\) in terms of sine and cosine: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\).
4. Step 4: Substitute the simplified expressions back into the LHS and simplify the fraction. Look for common terms that can be canceled out.
5. Step 5: Verify that the simplified LHS equals the right-hand side (RHS), \(2 \tan \theta\), by ensuring both sides are equivalent after simplification.

Key Concepts

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.

Secant and Tangent Functions

The secant function (sec θ) is the reciprocal of the cosine function, while the tangent function (tan θ) is the ratio of the sine function to the cosine function. These functions are fundamental in trigonometry and often appear in various identities and equations. Recognizing their relationships helps in manipulating and simplifying expressions involving them.

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves rearranging and simplifying expressions to prove identities. This includes factoring, expanding, and combining like terms. In trigonometry, it is essential to apply these techniques to transform one side of an equation into the other, thereby verifying the identity effectively.