Question

Which expression is equivalent to sqrt 1-cos(10x) / 2

Answer 1

The expression sqrt(1 - cos(10x) / 2) is equivalent to sin(5x), as derived from the half-angle identity for sine, which matches the given expression when theta is 5x.

Step-by-step explanation:

The expression sqrt(1 - cos(10x) / 2) is equivalent to sin(5x) due to the half-angle identity for sine, which states that sin^2(theta) = (1 - cos(2theta)) / 2. Hence, if we let theta be equal to 5x, we can rewrite the original expression as the square root of sin^2(5x), which is sin(5x). The radical sign indicates the principal (non-negative) square root, leading to the conclusion that the value inside the square root must also be non-negative, which is true for sin^2(theta).

Answer 2

The expression is

`\sqrt[]{(1-\cos (10x))/(2)}`

Using the half angle formula:

`\sin (\theta)/(2)=\pm\sqrt[]{(1-\cos \theta)/(2)}`

If θ=10x then we can replace it in the formula: