Reciprocal Identities Quiz: Fundamental Reciprocal Identities

  • 10th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) Which expression equals sinθ?

Explanation

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

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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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2) Simplify: sinθ·cscθ

Explanation

cscθ = 1/sinθ, so sinθ·cscθ = sinθ·(1/sinθ) = 1.

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3) Simplify: 1/cotθ

Explanation

cotθ = 1/tanθ, so 1/cotθ = tanθ.

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4) Express cosθ as a reciprocal.

Explanation

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

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5) Select all expressions equivalent to cscθ.

Explanation

cscθ = 1/sinθ. Also secθ/tanθ = (1/cosθ)/(sinθ/cosθ) = 1/sinθ = cscθ. Option E is cscθ itself.

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6) Simplify: 1/cscθ

Explanation

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

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7) Secθ equals 1/sinθ.

Explanation

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

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8) Simplify: secθ · cosθ

Explanation

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

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9) Simplify: secθ/cscθ

Explanation

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

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

Explanation

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

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11) Simplify: tanθ · cotθ

Explanation

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

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12) Select all expressions equivalent to 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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13) Simplify: (1/sinθ)·(1/cosθ)

Explanation

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

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14) Evaluate: sinθ·cscθ + cosθ·secθ

Explanation

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

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15) Simplify: 1/cscθ

Explanation

cscθ = 1/sinθ, so 1/cscθ = sinθ.

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16) Tanθ is the reciprocal of cotθ.

Explanation

cotθ = 1/tanθ, hence tanθ = 1/cotθ.

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17) Select all expressions equivalent to secθ.

Explanation

secθ is defined as 1/cosθ. Option B repeats secθ. 1/sinθ = cscθ, not secθ. cscθ ≠ secθ. sinθ/tanθ = sinθ/(sinθ/cosθ) = cosθ, which is not secθ.

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18) Cscθ is the reciprocal of sinθ.

Explanation

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

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19) Simplify: 1/secθ

Explanation

secθ = 1/cosθ, so 1/secθ = cosθ.

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20) Select all expressions that are identically equal to 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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Which expression equals sinθ?
Simplify: sinθ·cscθ
Simplify: 1/cotθ
Express cosθ as a reciprocal.
Select all expressions equivalent to cscθ.
Simplify: 1/cscθ
Secθ equals 1/sinθ.
Simplify: secθ · cosθ
Simplify: secθ/cscθ
Secθ·cscθ equals 1 for all θ in the domain.
Simplify: tanθ · cotθ
Select all expressions equivalent to tanθ.
Simplify: (1/sinθ)·(1/cosθ)
Evaluate: sinθ·cscθ + cosθ·secθ
Simplify: 1/cscθ
Tanθ is the reciprocal of cotθ.
Select all expressions equivalent to secθ.
Cscθ is the reciprocal of sinθ.
Simplify: 1/secθ
Select all expressions that are identically equal to 1.
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