Question

Which statement is true?
(1 point)
Responses

sine left parenthesis Start Fraction pi over 3 End Fraction right parenthesis equals cosine squared left parenthesis Start Fraction pi over 6 End Fraction right parenthesis minus sine squared left parenthesis Start Fraction pi over 6 End Fraction right parenthesis
Image with alt text: sine left parenthesis Start Fraction pi over 3 End Fraction right parenthesis equals cosine squared left parenthesis Start Fraction pi over 6 End Fraction right parenthesis minus sine squared left parenthesis Start Fraction pi over 6 End Fraction right parenthesis

tangent 21 degrees equals Start Fraction 2 tangent 42 degrees over 1 minus tangent squared 42 degrees End Fraction
Image with alt text: tangent 21 degrees equals Start Fraction 2 tangent 42 degrees over 1 minus tangent squared 42 degrees End Fraction

sine squared 36 degrees equals Start Fraction 1 minus cosine 72 degrees over 2 End Fraction
Image with alt text: sine squared 36 degrees equals Start Fraction 1 minus cosine 72 degrees over 2 End Fraction

sine left parenthesis Start Fraction pi over 6 End Fraction right parenthesis equals plus or minus Start Root Start Fraction 1 minus cosine left parenthesis Start Fraction pi over 12 End Fraction right parenthesis over 2 End Fraction End Root

Answer 1

Let's evaluate each statement to determine which is true.

1. **sine(π/3) = cosine²(π/6) - sine²(π/6)**
We know that:
- sin(π/3) = √3/2
- cos(π/6) = √3/2, so cos²(π/6) = (√3/2)² = 3/4
- sin(π/6) = 1/2, so sin²(π/6) = (1/2)² = 1/4
Therefore, the right side becomes:
cos²(π/6) - sin²(π/6) = 3/4 - 1/4 = 2/4 = 1/2, which does NOT equal sin(π/3) (√3/2).
**This statement is FALSE.**

2. **tangent(21 degrees) = (2 tan(42 degrees)) / (1 - tan²(42 degrees))**
This is a well-known double-angle formula for tangent, where:
tan(2θ) = (2 tan(θ)) / (1 - tan²(θ)). Here, θ = 21 degrees, hence 2θ = 42 degrees.
This is a true identity.
**This statement is TRUE.**

3. **sine²(36 degrees) = (1 - cos(72 degrees)) / 2**
We can use the identity: sin²(θ) = (1 - cos(2θ)) / 2. Here, θ = 36 degrees, then 2θ = 72 degrees. Therefore:
sin²(36 degrees) = (1 - cos(72 degrees)) / 2.
**This statement is TRUE.**

4. **sine(π/6) = ±√[(1 - cos(π/12)) / 2]**
We know that sin(π/6) = 1/2.
To evaluate the right side:
We also know that sin²(θ) = (1 - cos(2θ))/2, hence sin²(π/12) = (1 - cos(π/6))/2. Here, we would need to evaluate further. However, this does not check out because, for the right side, we would likely be looking for the value of sin(π/12) instead. Basically:
sin(π/6) = ±√[(1 - cos(π/12))/2] is not an immediate identity we can accept as is.
**This statement is FALSE.**

Final evaluation:
- The TRUE statements from those evaluated are 2 and 3. But if the question asks for a single true statement, and typically such questions emphasize the most direct trigonometric equality, **the second statement is the most direct and involved specifically with tangent functions**.