Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1

검증된 단계별 안내

Recognize that the expression \( \csc^2 t - 1 \) can be related to a fundamental trigonometric identity.

Recall the Pythagorean identity: \( \csc^2 t = 1 + \cot^2 t \).

Substitute \( \csc^2 t \) in the expression with \( 1 + \cot^2 t \) to get \( (1 + \cot^2 t) - 1 \).

Simplify the expression by subtracting 1: \( \cot^2 t \).

Conclude that the simplified expression is \( \cot^2 t \), which is a power of a trigonometric function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cosecant Function

The cosecant function, denoted as csc(t), is the reciprocal of the sine function. It is defined as csc(t) = 1/sin(t). Understanding this function is crucial for simplifying expressions involving csc²(t), as it relates directly to the sine function and its properties.

Pythagorean Identity

The Pythagorean identities are fundamental relationships in trigonometry that relate the squares of the sine and cosine functions. One key identity is sin²(t) + cos²(t) = 1. This identity can be rearranged to express csc²(t) in terms of sin²(t), which is essential for simplifying expressions like csc²(t) - 1.

Fundamental Trigonometric Identities

Fundamental trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where the functions are defined. These include reciprocal identities, Pythagorean identities, and co-function identities. Utilizing these identities allows for the simplification of complex trigonometric expressions into more manageable forms.

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.csc² t - 1