Reciprocal Identities Quiz: Fundamental Reciprocal Identities

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1) Cscθ is the reciprocal of sinθ.

True

False

By definition, cscθ = 1/sinθ, so it is the reciprocal of sinθ.

Explanation

By definition, cscθ = 1/sinθ, so it is the reciprocal of sinθ.

About This Quiz

Reciprocal Identities Quiz: Fundamental Reciprocal Identities - Quiz

Think you can spot reciprocal identities without hesitating? This quiz walks you through the key relationships between sine, cosine, tangent, and their reciprocals. You’ll practice quick conversions and see how these identities make bigger trig problems easier. Try the questions and build a stronger foundation in trigonometry.

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3) Simplify: tanθ · cotθ

Tanθ

Cotθ

Secθ

cotθ = 1/tanθ. Hence tanθ·cotθ = tanθ·(1/tanθ) = 1.

Explanation

cotθ = 1/tanθ. Hence tanθ·cotθ = tanθ·(1/tanθ) = 1.

4) Express cosθ as a reciprocal.

1/secθ

1/cscθ

1/cotθ

1/tanθ

Since secθ = 1/cosθ, invert both sides to get cosθ = 1/secθ.

Explanation

Since secθ = 1/cosθ, invert both sides to get cosθ = 1/secθ.

5) Tanθ is the reciprocal of cotθ.

True

False

cotθ = 1/tanθ, hence tanθ = 1/cotθ.

Explanation

cotθ = 1/tanθ, hence tanθ = 1/cotθ.

6) Simplify: 1/cscθ

Sinθ

Cosθ

Tanθ

Secθ

By definition, cscθ = 1/sinθ. Therefore 1/cscθ = sinθ.

Explanation

By definition, cscθ = 1/sinθ. Therefore 1/cscθ = sinθ.

7) Secθ equals 1/sinθ.

True

False

1/sinθ is cscθ, not secθ. secθ = 1/cosθ.

Explanation

1/sinθ is cscθ, not secθ. secθ = 1/cosθ.

8) Simplify: secθ · cosθ

Sinθ

Tanθ

Cscθ

secθ = 1/cosθ, so secθ·cosθ = (1/cosθ)·cosθ = 1.

Explanation

secθ = 1/cosθ, so secθ·cosθ = (1/cosθ)·cosθ = 1.

9) Evaluate: sinθ·cscθ + cosθ·secθ

Sinθ + cosθ

sinθ·cscθ = sinθ·(1/sinθ) = 1 and cosθ·secθ = cosθ·(1/cosθ) = 1, so the sum is 1 + 1 = 2.

Explanation

sinθ·cscθ = sinθ·(1/sinθ) = 1 and cosθ·secθ = cosθ·(1/cosθ) = 1, so the sum is 1 + 1 = 2.

10) Which expression equals sinθ?

1/cscθ

Cscθ

1/secθ

Cotθ

By the reciprocal identity, cscθ = 1/sinθ, so 1/cscθ = 1/(1/sinθ) = sinθ.

Explanation

By the reciprocal identity, cscθ = 1/sinθ, so 1/cscθ = 1/(1/sinθ) = sinθ.

11) Secθ·cscθ equals 1 for all θ in the domain.

True

False

secθ·cscθ = (1/cosθ)(1/sinθ) = 1/(sinθ cosθ), which is not identically 1.

Explanation

secθ·cscθ = (1/cosθ)(1/sinθ) = 1/(sinθ cosθ), which is not identically 1.

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16) Select all expressions equivalent to tanθ.

1/cotθ

Sinθ/cosθ

Secθ

Cotθ

Secθ/cscθ

tanθ is the reciprocal of cotθ, so 1/cotθ = tanθ. Also tanθ = sinθ/cosθ. Finally, secθ/cscθ = (1/cosθ)/(1/sinθ) = sinθ/cosθ = tanθ.

Explanation

tanθ is the reciprocal of cotθ, so 1/cotθ = tanθ. Also tanθ = sinθ/cosθ. Finally, secθ/cscθ = (1/cosθ)/(1/sinθ) = sinθ/cosθ = tanθ.

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17) Simplify: (1/sinθ)·(1/cosθ)

Cscθ + secθ

Cscθ·secθ

Tanθ

(1/sinθ)(1/cosθ) = 1/(sinθ cosθ) = cscθ·secθ since cscθ = 1/sinθ and secθ = 1/cosθ.

Explanation

(1/sinθ)(1/cosθ) = 1/(sinθ cosθ) = cscθ·secθ since cscθ = 1/sinθ and secθ = 1/cosθ.

18) Select all expressions that are identically equal to 1.

Sinθ·cscθ

Cosθ·secθ

Tanθ·cotθ

Secθ·cscθ

Cscθ/cscθ

sinθ·cscθ = sinθ·(1/sinθ) = 1; cosθ·secθ = cosθ·(1/cosθ) = 1; tanθ·cotθ = tanθ·(1/tanθ) = 1; cscθ/cscθ = 1. secθ·cscθ = (1/cosθ)(1/sinθ) = 1/(sinθ cosθ) ≠ 1.

Explanation

sinθ·cscθ = sinθ·(1/sinθ) = 1; cosθ·secθ = cosθ·(1/cosθ) = 1; tanθ·cotθ = tanθ·(1/tanθ) = 1; cscθ/cscθ = 1. secθ·cscθ = (1/cosθ)(1/sinθ) = 1/(sinθ cosθ) ≠ 1.

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