Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1/ tan² α + cot α tan α

Verified step by step guidance

1. Recognize that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \).
2. Rewrite \( \frac{1}{\tan^2 \alpha} \) as \( \cot^2 \alpha \) using the identity \( \cot \alpha = \frac{1}{\tan \alpha} \).
3. Substitute \( \cot^2 \alpha \) for \( \frac{1}{\tan^2 \alpha} \) in the expression.
4. Simplify \( \cot \alpha \tan \alpha \) to 1, since \( \cot \alpha = \frac{1}{\tan \alpha} \).
5. Combine the simplified terms: \( \cot^2 \alpha + 1 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, are essential for simplifying trigonometric expressions. Understanding these identities allows students to manipulate and transform expressions effectively.

Reciprocal Functions

Reciprocal functions in trigonometry refer to the relationships between sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. For example, the tangent function is the reciprocal of cotangent, and this relationship is crucial when simplifying expressions. Recognizing these relationships helps in rewriting expressions in a more manageable form.

Simplification Techniques

Simplification techniques in trigonometry involve rewriting complex expressions into simpler forms using identities and algebraic manipulation. This may include factoring, combining like terms, or substituting equivalent expressions. Mastering these techniques is vital for solving trigonometric equations and understanding their behavior in various contexts.