Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc² θ + sec² θ

검증된 단계별 안내

Recall the definitions of the cosecant and secant functions in terms of sine and cosine: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).

Rewrite the given expression \(\csc^{2} \theta + \sec^{2} \theta\) using these definitions: \(\left(\frac{1}{\sin \theta}\right)^{2} + \left(\frac{1}{\cos \theta}\right)^{2}\).

Simplify the expression to \(\frac{1}{\sin^{2} \theta} + \frac{1}{\cos^{2} \theta}\), which is a sum of two fractions.

Find a common denominator for the two fractions, which is \(\sin^{2} \theta \cos^{2} \theta\), and rewrite the expression as \(\frac{\cos^{2} \theta}{\sin^{2} \theta \cos^{2} \theta} + \frac{\sin^{2} \theta}{\sin^{2} \theta \cos^{2} \theta}\).

Combine the fractions into a single fraction: \(\frac{\cos^{2} \theta + \sin^{2} \theta}{\sin^{2} \theta \cos^{2} \theta}\), then use the Pythagorean identity \(\sin^{2} \theta + \cos^{2} \theta = 1\) to simplify the numerator.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Reciprocal Trigonometric Identities

Reciprocal identities express functions like cosecant and secant in terms of sine and cosine: csc θ = 1/sin θ and sec θ = 1/cos θ. These allow rewriting expressions involving csc² θ and sec² θ into fractions with sin θ and cos θ in the denominator.

Pythagorean Identities

Pythagorean identities relate sine and cosine functions, such as sin² θ + cos² θ = 1. These identities help simplify expressions by replacing squared terms or combining fractions to eliminate quotients and reduce complexity.

Algebraic Simplification of Trigonometric Expressions

After substituting reciprocal identities, algebraic techniques like finding common denominators and combining fractions are used to simplify the expression. The goal is to write the expression without quotients and only in terms of sin θ and cos θ.

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.csc² θ + sec² θ