Trigonometry Trigonometric Identities 2 — Questions and Answers
Question 1: Which identity correctly expresses cos(2θ) in terms of sin(θ) only?
- cos(2θ) = 1 − 2sin²(θ) (Correct answer)
- cos(2θ) = 2sin²(θ) − 1
- cos(2θ) = sin²(θ) − cos²(θ)
- cos(2θ) = 1 + 2sin²(θ)
Correct answer: cos(2θ) = 1 − 2sin²(θ)
The double-angle identity cos(2θ) = 1 − 2sin²(θ) is derived by substituting cos²θ = 1 − sin²θ into cos(2θ) = cos²θ − sin²θ.
Question 2: What is the simplified form of (1 − cos²θ) / sin θ?
- sin θ (Correct answer)
- cos θ
- tan θ
- cot θ
Correct answer: sin θ
Since 1 − cos²θ = sin²θ by the Pythagorean identity, dividing by sin θ gives sin θ.
Question 3: Which of the following equals sin(A + B)?
- sin A cos B + cos A sin B (Correct answer)
- sin A cos B − cos A sin B
- cos A cos B − sin A sin B
- cos A cos B + sin A sin B
Correct answer: sin A cos B + cos A sin B
The sum identity for sine is sin(A + B) = sin A cos B + cos A sin B.
Question 4: If tan θ = 3/4, what is the value of sin θ cos θ using identities?
- 12/25 (Correct answer)
- 7/25
- 3/5
- 4/5
Correct answer: 12/25
With tan θ = 3/4, we get sin θ = 3/5 and cos θ = 4/5 in a 3-4-5 triangle, so sin θ cos θ = (3/5)(4/5) = 12/25.
Question 5: Which identity is used to prove that cot²θ + 1 = csc²θ?
- Pythagorean identity sin²θ + cos²θ = 1 divided by sin²θ (Correct answer)
- Pythagorean identity sin²θ + cos²θ = 1 divided by cos²θ
- Reciprocal identity only
- Double-angle identity
Correct answer: Pythagorean identity sin²θ + cos²θ = 1 divided by sin²θ
Dividing sin²θ + cos²θ = 1 by sin²θ yields 1 + cot²θ = csc²θ, which rearranges to cot²θ + 1 = csc²θ.
Question 6: What is the value of sin(π/4) · cos(π/4)?
- 1/2 (Correct answer)
- √2/2
- √2/4
- 1
Correct answer: 1/2
sin(π/4) = cos(π/4) = √2/2, so their product is (√2/2)² = 2/4 = 1/2.
Question 7: Which expression is equivalent to (sec θ − 1)(sec θ + 1)?
- tan²θ (Correct answer)
- sin²θ
- cos²θ
- csc²θ
Correct answer: tan²θ
Expanding gives sec²θ − 1, and by the identity sec²θ − 1 = tan²θ.
Trigonometry Trigonometric Identities 2
Which identity correctly expresses cos(2θ) in terms of sin(θ) only?