Pythagorean Forms & Trig Identities Mixed Practice Quiz

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| Questions: 20 | Updated: Oct 31, 2025

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1) Which identity is equivalent to 1 + tan²(θ) = sec²(θ)?

Sin²(θ) + cos²(θ) = 1

1 + sin²(θ) = csc²(θ)

1 + cot²(θ) = csc²(θ)

Tan²(θ) = sec²(θ) − 1

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.

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

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.

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2) Simplify (sec²(θ) − 1)/(sec²(θ)).

Tan²(θ)

1 − cos²(θ)

1 − 1/sec²(θ)

Tan²(θ)/sec²(θ)

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).

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).

Tan²(θ)

Sec²(θ)

Given: csc²θ − cot²θ. Goal: Simplify.

Step 1: Substitute csc²θ = 1 + cot²θ.

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

So final answer is 1.

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).

4/3

5/4

5/3

3/5

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.

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.

−1

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.

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?

Csc(θ)

1 + 1/tan²(θ)

Sec²(θ)/tan²(θ)

1/cos²(θ)

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).

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).

√5

2√5/5

√5/2

5/√5

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.

Explanation

Submit

8) Choose the expression identically equal to 1:

Sec²(θ) − tan²(θ)

Csc²(θ) − cot²(θ) − 1

Tan²(θ) − sec²(θ)

1 − cot²(θ) − csc²(θ)

Given: sec²θ − tan²θ. Goal: Simplify.

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

So final answer is sec²(θ) − tan²(θ).

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).

169/144

144/169

169/25

25/144

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.

Explanation

Submit

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

They intersect only at x = 0.

They coincide on their shared domains.

Sec²(x) lies above.

1 + tan²(x) lies above.

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.

Explanation

Submit

11) Simplify (csc²(θ) − 1)/(cot²(θ)).

Csc²(θ)/cot²(θ)

(1 + cot²(θ) − 1)/cot(θ)

(1 + cot²(θ))/cot²(θ)

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

Step 1: Use identity csc²θ − 1 = cot²θ.

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

So final answer is 1.

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?

1 + tan²(θ) = sec²(θ)

1 + sec²(θ) = tan²(θ)

Tan²(θ) − 1 = sec²(θ)

Sec²(θ) + tan²(θ) = 1

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).

Explanation

Submit

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

1 − t²

1 + t²

T² − 1

1/(1 − t²)

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.

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)?

Cot²(θ) = csc²(θ) + 1

Cot²(θ) = csc²(θ) − 1

Csc²(θ) = 1 − cot²(θ)

1 − cot²(θ) = csc²(θ)

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.

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.

25/9

16/9

9/16

25/16

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.

Explanation

Submit

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

−48

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.

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)?

1 + x²

1 − x²

X² − 1

1/(1 − x²)

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.

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.

−1

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.

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).

5/13

12/13

13/12

13/5

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.

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).

1/tan²(θ) + 1 = sec²(θ)/tan²(θ)

1 + tan²(θ) = sec²(θ)

1 + cot²(θ) = csc²(θ)

Tan²(θ) + 1 = sec²(θ)

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).

Explanation

Submit

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