Quotient Identities Simplify & Verify Quiz

  • 11th Grade
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Cierra Henderson, MBA |
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 26 | Questions: 20 | Updated: Jan 22, 2026
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1) Simplify: (sin x)/(cos x)

Explanation

Given: sin/cos. Goal: simplify.

Step 1: By quotient identity, sin/cos = tan.

So, the final answer is tan x.

Submit
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About This Quiz
Quotient Identities Simplify & Verify Quiz - Quiz

Speed round: convert between tan, cot, sec, csc and sin/cos to simplify, verify identities, and evaluate at common angles. Expect reciprocal/quotient swaps, cancellations, and quick checks of “both sides defined” to keep proofs airtight.

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2) Simplify: (cos x)/(sin x)

Explanation

Given: cos/sin. Goal: simplify.

Step 1: cos/sin = cot.

So, the final answer is cot x.

Submit

3) Simplify: (sin x · csc x)/(cos x · sec x)

Explanation

Given: product ratio. Goal: simplify.

Step 1: sin·csc = 1; cos·sec = 1.

Step 2: 1/1 = 1.

So, the final answer is 1.

Submit

4) Which is true?

Explanation

Given: tan identities. Goal: select the true one.

Step 1: tan x = sin x / cos x.

So, the final answer is tan x = sin x / cos x.

Submit

5) Choose the equivalent expression to tan x:

Explanation

Given: tan. Goal: an equivalent form.

Step 1: tan x = sin x / cos x.

So, the final answer is sin x / cos x.

Submit

6) If tan θ = 3/4 and θ is acute, then sin θ / cos θ equals:

Explanation

Given: tanθ = 3/4. Goal: sin/cos.

Step 1: tan = sin/cos, so value is 3/4.

So, the final answer is 3/4.

Submit

7) Verify: (tan x)(cot x) equals

Explanation

Given: product tan·cot. Goal: evaluate.

Step 1: (sin/cos)(cos/sin) = 1.

So, the final answer is 1.

Submit

8) Which expression is equal to cot x?

Explanation

Given: cot. Goal: an equivalent form.

Step 1: cot x = 1/tan x (also = cos/sin).

So, the final answer is 1 / tan x.

Submit

9) Simplify: 1/(cot x)

Explanation

Given: reciprocal of cot. Goal: simplify.

Step 1: 1/cot x = tan x.

So, the final answer is tan x.

Submit

10) Simplify: (sin x)/(1/cot x)

Explanation

Given: sin divided by (1/cot). Goal: simplify.

Step 1: (sin)/(1/cot) = sin·cot = sin·(cos/sin) = cos.

So, the final answer is cos x.

Submit

11) Simplify tan x / sec x

Explanation

Given: tan/sec. Goal: simplify.

Step 1: (sin/cos)/(1/cos) = sin.

So, the final answer is sin x.

Submit

12) Simplify (cos x/sin x)·sin x

Explanation

Given: product. Goal: simplify.

Step 1: (cos/sin)·sin = cos.

So, the final answer is cos x.

Submit

13) Simplify: (csc x − cot x)(csc x + cot x)

Explanation

Given: conjugate product. Goal: simplify.

Step 1: (a−b)(a+b) = a² − b² → csc²x − cot²x.

Step 2: From 1 + cot²x = csc²x ⇒ csc²x − cot²x = 1.

So, the final answer is 1.

Submit

14) Which identity is true for all x where defined?

Explanation

Goal: pick a true identity.

Step 1: tan = 1/cot is always true where both are defined.

So, the final answer is tan x = 1/(cot x).

Submit

15) Verify the identity: tan x · cos x = ?

Explanation

Given: tan·cos. Goal: evaluate.

Step 1: tan = sin/cos → (sin/cos)·cos = sin.

So, the final answer is sin x.

Submit

16) Simplify: (sec x)/(csc x)

Explanation

Given: sec/csc. Goal: simplify.

Step 1: (1/cos)/(1/sin) = sin/cos = tan.

So, the final answer is tan x.

Submit

17) Simplify: (cot x)/(csc x)

Explanation

Given: cot/csc. Goal: simplify.

Step 1: (cos/sin)/(1/sin) = cos.

So, the final answer is cos x.

Submit

18) Evaluate using quotient identity: If x = 30°, tan x equals

Explanation

Given: x = 30°. Goal: tan 30°.

Step 1: tan 30° = 1/√3 = √3/3.

So, the final answer is √3/3.

Submit

19) Simplify sin x/cos x + cos x/sin x

Explanation

Given: sum of quotients. Goal: simplify.

Step 1: Common denominator sinx·cosx → (sin²x + cos²x)/(sinx·cosx).

Step 2: sin²x + cos²x = 1.

So, the final answer is 1/(sin x cos x).

Submit

20) Simplify: (sin² x)/(sin x · cos x)

Explanation

Given: sin²/(sin·cos). Goal: simplify.

Step 1: Cancel sin → sin/cos = tan.

So, the final answer is tan x.

Submit
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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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Simplify: (sin x)/(cos x)
Simplify: (cos x)/(sin x)
Simplify: (sin x · csc x)/(cos x · sec x)
Which is true?
Choose the equivalent expression to tan x:
If tan θ = 3/4 and θ is acute, then sin θ / cos θ equals:
Verify: (tan x)(cot x) equals
Which expression is equal to cot x?
Simplify: 1/(cot x)
Simplify: (sin x)/(1/cot x)
Simplify tan x / sec x
Simplify (cos x/sin x)·sin x
Simplify: (csc x − cot x)(csc x + cot x)
Which identity is true for all x where defined?
Verify the identity: tan x · cos x = ?
Simplify: (sec x)/(csc x)
Simplify: (cot x)/(csc x)
Evaluate using quotient identity: If x = 30°, tan x equals
Simplify sin x/cos x + cos x/sin x
Simplify: (sin² x)/(sin x · cos x)
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