Quotient Identities Quiz: Fundamental Quotient Identities

  • 10th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) Cotθ equals sinθ divided by cosθ.

Explanation

cotθ = cosθ/sinθ, not sinθ/cosθ. It is the reciprocal of tanθ.

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About This Quiz
Quotient Identities Quiz: Fundamental Quotient Identities - Quiz

Curious why tangent and cotangent can be expressed through sine and cosine? In this quiz, you’ll break down the fundamental quotient identities and see exactly how they arise from right-triangle ratios and unit-circle coordinates. You’ll practice rewriting expressions, interpreting graphs, and applying the identities to simplify problems. As you work,... see moreyou’ll gain clarity about how these relationships help streamline trigonometric reasoning, giving you a dependable framework for tackling more complex expressions and equations.
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2) Simplify: cosθ/sinθ

Explanation

By definition, cotθ = cosθ/sinθ for sinθ ≠ 0.

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3) Tanθ·cotθ equals 1 whenever both are defined.

Explanation

tanθ·cotθ = (sinθ/cosθ)·(cosθ/sinθ) = 1, for sinθ ≠ 0 and cosθ ≠ 0.

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4) Select all expressions equivalent to tanθ.

Explanation

tanθ = sinθ/cosθ. Since cotθ = cosθ/sinθ, 1/cotθ = sinθ/cosθ = tanθ. Also secθ/cscθ = (1/cosθ)/(1/sinθ) = sinθ/cosθ = tanθ. adjacent/opposite is cotθ, not tanθ.

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5) Simplify: cotθ·sinθ

Explanation

cotθ = cosθ/sinθ, so cotθ·sinθ = (cosθ/sinθ)·sinθ = cosθ.

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6) When cosθ = 0, which expressions are undefined?

Explanation

If cosθ = 0, tanθ = sinθ/cosθ and sinθ/cosθ are undefined due to division by zero. secθ = 1/cosθ is also undefined. cotθ = cosθ/sinθ equals 0/sinθ = 0 when sinθ ≠ 0, so defined.

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7) If sinθ = 3/5 and cosθ = 4/5 (acute θ), find tanθ.

Explanation

tanθ = sinθ/cosθ = (3/5)/(4/5) = 3/4.

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8) Simplify: tanθ ÷ (sinθ/cosθ)

Explanation

Since tanθ = sinθ/cosθ, the quotient is (sinθ/cosθ)/(sinθ/cosθ) = 1, for sinθ ≠ 0 and cosθ ≠ 0.

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9) Write tanθ as a quotient of sine and cosine.

Explanation

By the quotient identity, tanθ = sinθ/cosθ for all θ with cosθ ≠ 0.

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10) Simplify: sinθ/cosθ

Explanation

By definition, tanθ = sinθ/cosθ for cosθ ≠ 0.

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11) Select all identities that are always true (within domains).

Explanation

A: tanθ = sinθ/cosθ by definition. B: cotθ = cosθ/sinθ by definition. C: 1/(cosθ/sinθ) = sinθ/cosθ = tanθ. D: secθ/cscθ = (1/cosθ)/(1/sinθ) = sinθ/cosθ = tanθ. E is incorrect since sinθ·cosθ ≠ tanθ.

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12) Tanθ equals secθ divided by sinθ.

Explanation

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

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13) Simplify: tanθ·cosθ

Explanation

tanθ = sinθ/cosθ, so tanθ·cosθ = (sinθ/cosθ)·cosθ = sinθ.

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14) Tanθ equals sinθ divided by cosθ.

Explanation

This is the fundamental quotient identity: tanθ = sinθ/cosθ, defined when cosθ ≠ 0.

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15) Select all expressions equivalent to cotθ.

Explanation

cotθ = cosθ/sinθ = 1/tanθ. On the unit circle, x = cosθ and y = sinθ, so x/y = cosθ/sinθ = cotθ. adjacent/opposite is the triangle ratio for cotθ. sinθ/cosθ equals tanθ, not cotθ.

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16) Simplify: (sinθ/cosθ)·(cosθ/sinθ)

Explanation

Multiply numerators and denominators: (sinθ·cosθ)/(cosθ·sinθ) = 1, provided sinθ and cosθ are nonzero.

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17) Simplify: (sin^2θ)/(sinθ·cosθ)

Explanation

(sin^2θ)/(sinθ·cosθ) = [sinθ·sinθ]/[sinθ·cosθ] = sinθ/cosθ = tanθ (sinθ ≠ 0, cosθ ≠ 0).

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18) Simplify: (cos^2θ)/(sinθ·cosθ)

Explanation

(cos^2θ)/(sinθ·cosθ) = [cosθ·cosθ]/[sinθ·cosθ] = cosθ/sinθ = cotθ (sinθ ≠ 0, cosθ ≠ 0).

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19) Tan(−θ) = −tanθ follows from tanθ = sinθ/cosθ and the parity of sine and cosine.

Explanation

sin(−θ) = −sinθ (odd) and cos(−θ) = cosθ (even), so tan(−θ) = sin(−θ)/cos(−θ) = (−sinθ)/cosθ = −tanθ.

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20) If sinθ = 3/5 and cosθ = 4/5 (acute θ), find cotθ.

Explanation

cotθ = cosθ/sinθ = (4/5)/(3/5) = 4/3.

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Cotθ equals sinθ divided by cosθ.
Simplify: cosθ/sinθ
Tanθ·cotθ equals 1 whenever both are defined.
Select all expressions equivalent to tanθ.
Simplify: cotθ·sinθ
When cosθ = 0, which expressions are undefined?
If sinθ = 3/5 and cosθ = 4/5 (acute θ), find tanθ.
Simplify: tanθ ÷ (sinθ/cosθ)
Write tanθ as a quotient of sine and cosine.
Simplify: sinθ/cosθ
Select all identities that are always true (within domains).
Tanθ equals secθ divided by sinθ.
Simplify: tanθ·cosθ
Tanθ equals sinθ divided by cosθ.
Select all expressions equivalent to cotθ.
Simplify: (sinθ/cosθ)·(cosθ/sinθ)
Simplify: (sin^2θ)/(sinθ·cosθ)
Simplify: (cos^2θ)/(sinθ·cosθ)
Tan(−θ) = −tanθ follows from tanθ = sinθ/cosθ and the parity...
If sinθ = 3/5 and cosθ = 4/5 (acute θ), find cotθ.
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