Perform each indicated operation and simplify the result so that there are no quotients.
(1 + tan θ)² - 2 tan θ

Verified step by step guidance

1. Start by expanding the expression \((1 + \tan \theta)^2\). This is a binomial square, so use the formula \((a + b)^2 = a^2 + 2ab + b^2\).
2. Apply the formula: \((1 + \tan \theta)^2 = 1^2 + 2 \cdot 1 \cdot \tan \theta + (\tan \theta)^2\).
3. Simplify the expression: \(1 + 2\tan \theta + \tan^2 \theta\).
4. Subtract \(2\tan \theta\) from the expanded expression: \(1 + 2\tan \theta + \tan^2 \theta - 2\tan \theta\).
5. Combine like terms: \(1 + \tan^2 \theta\).

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the properties of the tangent function is essential for manipulating expressions involving it.

Algebraic Expansion

Algebraic expansion involves applying the distributive property to simplify expressions. In the context of the given expression (1 + tan θ)², this means expanding it to (1 + tan θ)(1 + tan θ), which results in 1 + 2tan θ + tan² θ. Mastery of expansion techniques is crucial for simplifying complex trigonometric expressions.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves reducing them to a more manageable form, often eliminating fractions or quotients. This can include combining like terms, factoring, or using trigonometric identities. In the given problem, simplifying the result after performing the indicated operations is key to achieving a final expression without quotients.