Pythagorean Forms & Trig Identities Mixed Practice Quiz

  • 12th Grade
Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7387 | Total Attempts: 9,541,629
| Questions: 20 | Updated: Dec 11, 2025
Please wait...
Question 1 / 20
0 %
0/100
Score 0/100
1) Which identity is equivalent to 1 + tan²(θ) = sec²(θ)?

Explanation

Given: 1 + tan²θ = sec²θ. Goal: Write an equivalent form.

Step 1: Subtract 1 from both sides tan²θ = sec²θ − 1.

So final answer is tan²(θ) = sec²(θ) − 1.

Submit
Please wait...
About This Quiz
Pythagorean Forms & Trig Identities Mixed Practice Quiz - Quiz

Put it all together—proofs, simplifications, and quick evaluations using the Pythagorean family. You’ll flip between tan/sec and cot/csc forms, verify identities, and use triangles or the unit circle to pin down exact values. A crisp workout for both theory and skills.

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) Simplify (sec²(θ) − 1)/(sec²(θ)).

Explanation

Given: (sec²θ − 1)/(sec²θ). Goal: Simplify the expression.

Step 1: Use identity sec²θ − 1 = tan²θ.

Step 2: Substitute (tan²θ)/(sec²θ).

So final answer is tan²(θ)/sec²(θ).

Submit
3) Using 1 + cot²(θ) = csc²(θ), simplify csc²(θ) − cot²(θ).

Explanation

Given: csc²θ − cot²θ. Goal: Simplify.

Step 1: Substitute csc²θ = 1 + cot²θ.

Step 2: Simplify (1 + cot²θ) − cot²θ = 1.

So final answer is 1.

Submit
4) Let tan(θ) = 3/4 with θ in Quadrant I. Compute sec(θ).

Explanation

Given: tanθ = 3/4, θ in QI. Goal: Find secθ.

Step 1: Use triangle opp = 3, adj = 4, hyp = 5.

Step 2: cosθ = adj/hyp = 4/5.

Step 3: secθ = 1/cosθ = 5/4.

So final answer is 5/4.

Submit
5) If sec²(θ) − tan²(θ) = k for all θ where defined, find k.

Explanation

Given: sec²θ − tan²θ. Goal: Simplify to find k.

Step 1: From identity, 1 + tan²θ = sec²θ.

Step 2: Rearranging sec²θ − tan²θ = 1.

So final answer is 1.

Submit
6) Verify: For any θ where defined, (1 + tan²(θ))/tan²(θ) equals which expression?

Explanation

Given: (1 + tan²θ)/tan²θ. Goal: Simplify using identity.

Step 1: Replace numerator 1 + tan²θ = sec²θ.

Step 2: Expression becomes sec²θ/tan²θ.

So final answer is sec²(θ)/tan²(θ).

Submit
7) Given cot(θ) = 2 and θ in Quadrant I, find csc(θ).

Explanation

Given: cotθ = 2, θ in QI. Goal: Find cscθ.

Step 1: cot = adj/opp = 2/1 ⇒ hyp = √(2² + 1²) = √5.

Step 2: sinθ = opp/hyp = 1/√5.

Step 3: cscθ = 1/sinθ = √5.

So final answer is √5.

Submit
8) Choose the expression identically equal to 1:

Explanation

Given: sec²θ − tan²θ. Goal: Simplify.

Step 1: From Pythagorean identity sec²θ − tan²θ = 1.

So final answer is sec²(θ) − tan²(θ).

Submit
9) If sin(θ) = 5/13 and θ in Quadrant II, compute 1 + cot²(θ).

Explanation

Given: sinθ = 5/13, θ in QII. Goal: Find 1 + cot²θ.

Step 1: cos²θ = 1 − (5/13)² = 144/169 cosθ = −12/13.

Step 2: cotθ = cosθ/sinθ = (−12/13)/(5/13) = −12/5.

Step 3: 1 + cot²θ = 1 + (144/25) = 169/25.

So final answer is 169/25.

Submit
10) A graph shows y = sec²(x) and y = 1 + tan²(x). Which statement is true?

Explanation

Given: y = sec²x and y = 1 + tan²x. Goal: Compare both.

Step 1: Identity 1 + tan²x = sec²x holds true for all x where defined.

Step 2: Therefore, both graphs coincide on their shared domains.

So final answer is They coincide on their shared domains.

Submit
11) Simplify (csc²(θ) − 1)/(cot²(θ)).

Explanation

Given: (csc²θ − 1)/(cot²θ). Goal: Simplify.

Step 1: Use identity csc²θ − 1 = cot²θ.

Step 2: Substitute (cot²θ)/(cot²θ) = 1.

So final answer is 1.

Submit
12) If sec(θ) = k and tan(θ) = √(k² − 1), which identity relates these?

Explanation

Given: secθ = k, tanθ = √(k² − 1). Goal: Find the connecting identity.

Step 1: Substitute into 1 + tan²θ 1 + (k² − 1) = k².

Step 2: Since sec²θ = k², identity holds.

So final answer is 1 + tan²(θ) = sec²(θ).

Submit
13) Let θ satisfy tan(θ) = t. Express sec²(θ) in terms of t.

Explanation

Given: tanθ = t. Goal: Find sec²θ in terms of t.

Step 1: From identity 1 + tan²θ = sec²θ.

Step 2: Substitute sec²θ = 1 + t².

So final answer is 1 + t².

Submit
14) Which is a correct rearrangement of 1 + cot²(θ) = csc²(θ)?

Explanation

Given: 1 + cot²θ = csc²θ. Goal: Rearrange for cot²θ.

Step 1: Subtract 1 from both sides cot²θ = csc²θ − 1.

So final answer is cot²(θ) = csc²(θ) − 1.

Submit
15) Evaluate sec²(θ)/tan²(θ) when sin(θ) = 3/5 and θ in Quadrant II.

Explanation

Given: sinθ = 3/5, θ in QII. Goal: Find sec²θ/tan²θ.

Step 1: cos²θ = 1 − (3/5)² = 16/25 cosθ = −4/5.

Step 2: tanθ = sinθ/cosθ = (3/5)/(−4/5) = −3/4 tan²θ = 9/16.

Step 3: sec²θ = 1/cos²θ = 25/16.

Step 4: sec²θ/tan²θ = (25/16)/(9/16) = 25/9.

So final answer is 25/9.

Submit
16) If tan(θ) = 7, compute csc²(θ) − cot²(θ).

Explanation

Given: tanθ = 7. Goal: Simplify csc²θ − cot²θ.

Step 1: From identity 1 + cot²θ = csc²θ.

Step 2: Subtract cot²θ csc²θ − cot²θ = 1.

So final answer is 1.

Submit
17) Given cot(θ) = x and x > 0, which expression equals csc²(θ)?

Explanation

Given: cotθ = x. Goal: Express csc²θ.

Step 1: Use identity 1 + cot²θ = csc²θ.

Step 2: Substitute cotθ = x csc²θ = 1 + x².

So final answer is 1 + x².

Submit
18) Consider (sec²(θ) − 1)/(tan²(θ)) = k. Find k.

Explanation

Given: (sec²θ − 1)/(tan²θ). Goal: Simplify to find k.

Step 1: sec²θ − 1 = tan²θ.

Step 2: Substitute (tan²θ)/(tan²θ) = 1.

So final answer is 1.

Submit
19) If θ is acute and tan(θ) = 12/5, find sin(θ).

Explanation

Given: tanθ = 12/5, θ acute. Goal: Find sinθ.

Step 1: opp = 12, adj = 5 ⇒ hyp = 13.

Step 2: sinθ = opp/hyp = 12/13.

So final answer is 12/13.

Submit
20) Starting from 1 + tan²(θ) = sec²(θ), divide both sides by tan²(θ).

Explanation

Given: 1 + tan²θ = sec²θ. Goal: Divide both sides by tan²θ.

Step 1: (1 + tan²θ)/tan²θ = sec²θ/tan²θ.

Step 2: Expand 1/tan²θ + 1 = sec²θ/tan²θ.

So final answer is 1/tan²(θ) + 1 = sec²(θ)/tan²(θ).

Submit
×
Saved
Thank you for your feedback!
View My Results
Cancel
  • All
    All (20)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Which identity is equivalent to 1 + tan²(θ) =...
Simplify (sec²(θ) − 1)/(sec²(θ)).
Using 1 + cot²(θ) = csc²(θ), simplify...
Let tan(θ) = 3/4 with θ in Quadrant I. Compute...
If sec²(θ) − tan²(θ) = k for all θ...
Verify: For any θ where defined, (1 +...
Given cot(θ) = 2 and θ in Quadrant I, find csc(θ).
Choose the expression identically equal to 1:
If sin(θ) = 5/13 and θ in Quadrant II, compute 1 +...
A graph shows y = sec²(x) and y = 1 + tan²(x). Which...
Simplify (csc²(θ) − 1)/(cot²(θ)).
If sec(θ) = k and tan(θ) = √(k² − 1),...
Let θ satisfy tan(θ) = t. Express sec²(θ) in...
Which is a correct rearrangement of 1 + cot²(θ) =...
Evaluate sec²(θ)/tan²(θ) when sin(θ) = 3/5...
If tan(θ) = 7, compute csc²(θ) −...
Given cot(θ) = x and x > 0, which expression equals...
Consider (sec²(θ) − 1)/(tan²(θ)) = k. Find...
If θ is acute and tan(θ) = 12/5, find sin(θ).
Starting from 1 + tan²(θ) = sec²(θ), divide both...
Alert!

Advertisement