Reciprocal Identities Proof Quiz (HSF-TF.C.9)

1) Which reciprocal identity is correct?

Tan(θ) = 1/cos(θ)

Cos(θ) = 1/sec(θ)

Sec(θ) = 1/sin(θ)

Csc(θ) = 1/tan(θ)

Given: Reciprocal identities. Goal: Identify the correct one. Step 1: Recall cosθ · secθ = 1 ⇒ cosθ = 1/secθ. So final answer is cos(θ) = 1/sec(θ).

Explanation

Given: Reciprocal identities. Goal: Identify the correct one. Step 1: Recall cosθ · secθ = 1 ⇒ cosθ = 1/secθ. So final answer is cos(θ) = 1/sec(θ).

2) Simplify completely: sin(θ)·csc(θ) + cos(θ)·sec(θ)

0

1

2

Sin(θ) + cos(θ)

Given: sin·csc + cos·sec. Step 1: sin·csc = 1. Step 2: cos·sec = 1. Step 3: Sum = 2. So final answer is 2.

Explanation

Given: sin·csc + cos·sec. Step 1: sin·csc = 1. Step 2: cos·sec = 1. Step 3: Sum = 2. So final answer is 2.

3) Rewrite using only sine and cosine: cot(θ)

Sin(θ)/cos(θ)

Cos(θ)/sin(θ)

1/(sin(θ)cos(θ))

Sin²(θ)

Given: cotθ. Step 1: cotθ = cosθ/sinθ. So final answer is cos(θ)/sin(θ).

Explanation

Given: cotθ. Step 1: cotθ = cosθ/sinθ. So final answer is cos(θ)/sin(θ).

4) If csc(α) = 13/5 and α is acute, find sin(α).

5/13

12/13

13/5

5/12

Given: cscα = 13/5, α acute. Step 1: cscα = 1/sinα ⇒ sinα = 5/13. So final answer is 5/13.

Explanation

Given: cscα = 13/5, α acute. Step 1: cscα = 1/sinα ⇒ sinα = 5/13. So final answer is 5/13.

5) Decide if the identity holds for all θ where defined: csc(θ) − sin(θ) = (1 − sin²(θ))/sin(θ)

True; equals cos²(θ)/sin(θ)

True; equals 1/sin(θ)

False; RHS equals cos(θ)/sin(θ)

False; LHS equals 1/sin(θ) − sin(θ) = (1 − sin²(θ))/sin(θ) + sin(θ)

Given: LHS = 1/sin − sin. Step 1: 1/sin − sin = (1 − sin²)/sin. Step 2: 1 − sin² = cos² ⇒ RHS = cos²/sin.

Explanation

Given: LHS = 1/sin − sin. Step 1: 1/sin − sin = (1 − sin²)/sin. Step 2: 1 − sin² = cos² ⇒ RHS = cos²/sin.

6) Simplify (1 + tan²(θ))·cos²(θ)

Tan²(θ)

1

Sec²(θ)

Sin²(θ)

Given: (1 + tan²θ)cos²θ. Step 1: 1 + tan²θ = sec²θ. Step 2: sec²θ · cos²θ = 1. So final answer is 1.

Explanation

Given: (1 + tan²θ)cos²θ. Step 1: 1 + tan²θ = sec²θ. Step 2: sec²θ · cos²θ = 1. So final answer is 1.

7) If sec(x) = 5/4 and x is acute, compute cos(x).

4/5

5/4

3/5

1/5

Given: sec x = 5/4. Step 1: cos x = 1/sec x = 4/5. So final answer is 4/5.

Explanation

Given: sec x = 5/4. Step 1: cos x = 1/sec x = 4/5. So final answer is 4/5.

8) Simplify to a single trig function: sin(θ)/(1 − cos(θ))

Tan(θ/2)

Csc(θ) + cot(θ)

Sec(θ) + tan(θ)

1 − cos(θ)

Given: sin/(1 − cos). Step 1: Multiply by (1 + cos)/(1 + cos). Step 2: Simplify to (1 + cos)/sin = cscθ + cotθ.

Explanation

Given: sin/(1 − cos). Step 1: Multiply by (1 + cos)/(1 + cos). Step 2: Simplify to (1 + cos)/sin = cscθ + cotθ.

9) Evaluate exactly: csc(π/6) + sec(π/3)

2

2.5

4

2 + 2/√3

Given: csc(π/6)=2, sec(π/3)=2. Step 3: Sum = 4.

Explanation

Given: csc(π/6)=2, sec(π/3)=2. Step 3: Sum = 4.

10) Which is always true?

Secθ − tanθ = 1/(secθ + tanθ)

Cscθ − sinθ = 1

Cotθ = sinθ·cscθ

Tanθ = secθ/sinθ

Given: (sec − tan)(sec + tan) = sec² − tan² = 1 ⇒ sec − tan = 1/(sec + tan).

Explanation

Given: (sec − tan)(sec + tan) = sec² − tan² = 1 ⇒ sec − tan = 1/(sec + tan).

11) Simplify: (sec(θ) − cos(θ))/sin²(θ)

Tanθ·cscθ

Cscθ·cotθ

Secθ·tanθ

1/(1 − cosθ)

Given: (1/cos − cos)/sin². Simplify to 1/cos = secθ = tanθ·cscθ.

Explanation

Given: (1/cos − cos)/sin². Simplify to 1/cos = secθ = tanθ·cscθ.

12) Which is correct?

Sec(x) = csc(π/2 − x)

Csc(x) = sec(π − x)

Tan(x) = cot(π/2 + x)

Cot(x) = tan(π − x)

Correct is: sec x = csc(π/2 − x) because this is a co-function identity, which states that a trigonometric function of an angle is equal to its co-function of the complementary angle.

Explanation

Correct is: sec x = csc(π/2 − x) because this is a co-function identity, which states that a trigonometric function of an angle is equal to its co-function of the complementary angle.

13) Right triangle with angle θ, adjacent = 9, hypotenuse = 15. Compute sec(θ) − csc(θ).

5/3 − 3/5

3/5 − 5/3

15/9 − 15/12

9/15 − 12/15

Step 1: Given adjacent = 9 and hypotenuse = 15.

cos(θ) = adjacent / hypotenuse = 9 / 15.

So, sec(θ) = 1 / cos(θ) = 15 / 9.

Step 2: Find the opposite side using Pythagoras’ theorem.

opposite = √(15² − 9²) = √(225 − 81) = √144 = 12.

Step 3: sin(θ) = opposite / hypotenuse = 12 / 15.

So, csc(θ) = 1 / sin(θ) = 15 / 12.

Step 4: Compute sec(θ) − csc(θ).

sec(θ) − csc(θ) = 15/9 − 15/12.

So, the final answer is 15/9 − 15/12.

Explanation

Step 1: Given adjacent = 9 and hypotenuse = 15.

cos(θ) = adjacent / hypotenuse = 9 / 15.

So, sec(θ) = 1 / cos(θ) = 15 / 9.

Step 2: Find the opposite side using Pythagoras’ theorem.

opposite = √(15² − 9²) = √(225 − 81) = √144 = 12.

Step 3: sin(θ) = opposite / hypotenuse = 12 / 15.

So, csc(θ) = 1 / sin(θ) = 15 / 12.

Step 4: Compute sec(θ) − csc(θ).

sec(θ) − csc(θ) = 15/9 − 15/12.

So, the final answer is 15/9 − 15/12.

14) Simplify completely: (cscθ + cotθ)² − (cscθ − cotθ)²

4 cscθ·cotθ

4 cscθ

4 cotθ

0

Step 1: Use the difference of squares: (A + B)² − (A − B)² = 4AB.

Step 2: Let A = csc(θ) and B = cot(θ) ⇒ expression = 4·csc(θ)·cot(θ).

So, the final answer is 4 csc(θ)·cot(θ).

Explanation

Step 1: Use the difference of squares: (A + B)² − (A − B)² = 4AB.

Step 2: Let A = csc(θ) and B = cot(θ) ⇒ expression = 4·csc(θ)·cot(θ).

So, the final answer is 4 csc(θ)·cot(θ).

15) Determine if true for all θ where defined: cot(θ) = cos(θ)·csc(θ)

True

False; equals sin(θ)·sec(θ)

False; equals sec(θ)·csc(θ)

False; equals cos(θ)·sec(θ)

Step 1: cot(θ) = cos(θ)/sin(θ).

Step 2: cos(θ)·csc(θ) = cos(θ)·(1/sin(θ)) = cos(θ)/sin(θ) = cot(θ).

So, the final answer is True.

Explanation

Step 1: cot(θ) = cos(θ)/sin(θ).

Step 2: cos(θ)·csc(θ) = cos(θ)·(1/sin(θ)) = cos(θ)/sin(θ) = cot(θ).

So, the final answer is True.

16) Simplify to a single function: (1 − cos(θ))/sin(θ)

Tan(θ/2)

Cscθ − tanθ

Secθ − tanθ

Sinθ/(1 − cosθ)

Step 1: Use the half-angle identity: tan(θ/2) = (1 − cosθ)/sinθ (where defined).

Step 2: Therefore, (1 − cosθ)/sinθ = tan(θ/2).

So, the final answer is tan(θ/2).

Explanation

Step 1: Use the half-angle identity: tan(θ/2) = (1 − cosθ)/sinθ (where defined).

Step 2: Therefore, (1 − cosθ)/sinθ = tan(θ/2).

So, the final answer is tan(θ/2).

17) Evaluate exactly: sin(π/4)·csc(π/4) − cos(π/6)·sec(π/6)

0

1

2

3

Step 1: sin(π/4) = √2/2 and csc(π/4) = √2 ⇒ product = 1.

Step 2: cos(π/6) = √3/2 and sec(π/6) = 2/√3 ⇒ product = 1.

Step 3: Difference = 1 − 1 = 0.

So, the final answer is 0.

Explanation

Step 1: sin(π/4) = √2/2 and csc(π/4) = √2 ⇒ product = 1.

Step 2: cos(π/6) = √3/2 and sec(π/6) = 2/√3 ⇒ product = 1.

Step 3: Difference = 1 − 1 = 0.

So, the final answer is 0.

18) Simplify: (csc(θ) + sec(θ))·(sin(θ) + cos(θ))

2

Secθ + cscθ

Tanθ + cotθ

2 + tanθ + cotθ

Step 1: Expand: csc·sin + csc·cos + sec·sin + sec·cos.

Step 2: csc·sin = 1 and sec·cos = 1; csc·cos = cot and sec·sin = tan.

Step 3: Sum = 1 + cot + tan + 1 = 2 + tanθ + cotθ.

So, the final answer is 2 + tanθ + cotθ.

Explanation

Step 1: Expand: csc·sin + csc·cos + sec·sin + sec·cos.

Step 2: csc·sin = 1 and sec·cos = 1; csc·cos = cot and sec·sin = tan.

Step 3: Sum = 1 + cot + tan + 1 = 2 + tanθ + cotθ.

So, the final answer is 2 + tanθ + cotθ.

19) Is csc²(θ) − cot²(θ) = 1 an identity?

Yes; follows from 1 + cot²(θ) = csc²(θ)

No; only true at θ = 0

No; equals cos²(θ)

Yes; equals sec²(θ)

Step 1: Start with the Pythagorean identity: csc²θ = 1 + cot²θ.

Step 2: Rearrange: csc²θ − cot²θ = 1.

So, the final answer is Yes; follows from 1 + cot²(θ) = csc²(θ).

Explanation

Step 1: Start with the Pythagorean identity: csc²θ = 1 + cot²θ.

Step 2: Rearrange: csc²θ − cot²θ = 1.

So, the final answer is Yes; follows from 1 + cot²(θ) = csc²(θ).