Question

Step to derive cos(x/2)= -root1 cosx/2 provide the justifications for steps in the derivation shown. Double the angle identity simplifying.

Answer 1

To derive the cosine half-angle identity, the double-angle cosine identity is rearranged and then the angle θ is replaced with x/2. Finally, the square root is taken with a ± sign to account for the different quadrants the angle may reside in.

Step-by-step explanation:

To derive the half-angle identity cos(x/2) = ±√(1 + cos(x))/2, we start from the cos(2θ) identities. We can write cos(2θ) in three different ways:

• cos(2θ) = cos²θ - sin²θ
• cos(2θ) = 2cos²θ - 1
• cos(2θ) = 1 - 2sin²θ

We are interested in the form involving only cosine, so we use the second one:

1. cos(2θ) = 2cos²θ - 1
2. Rearrange to solve for cos²θ: cos²θ = (1 + cos(2θ))/2
3. Replace θ with x/2 to get cos²(x/2) = (1 + cos(x))/2
4. Take the square root of both sides, remembering the ± sign to indicate that the cosine function can be positive or negative depending on the quadrant: cos(x/2) = ±√(1 + cos(x))/2

Thus, we find the desired half-angle identity for cosine. The ± sign accounts for the different quadrants where the angle x/2 may lie, indicating that the cosine of that angle may be either positive or negative.