Use the half-angle formula to simplify the given expression.
squareroot 1+cos12x/2
did you even look up your half-angle formula? It says that
cos(θ/2) = √[(1+cosθ)/2]
I think the substitution is now clear.
is it cos12x?
No.
You are half right. θ=12x is the right substitution to use, but if θ = 12x, what you have is cos(θ/2) = cos 6x
To simplify the given expression using the half-angle formula, we will start by applying the formula and then simplifying the expression accordingly. The half-angle formula for cosine is:
cos(x/2) = ±√((1 + cos(x))/2)
Let's use this formula to simplify the expression sqrt(1 + cos(12x)/2):
1. Start with the given expression: sqrt(1 + cos(12x)/2).
2. Apply the half-angle formula for cosine: cos(12x/2) = ±√((1 + cos(12x))/2).
3. Substitute cos(12x) into the formula: cos(12x/2) = ±√((1 + cos(12x))/2).
4. Simplify the expression inside the square root: sqrt(1 + cos(12x))
5. Combine the square root and the ± sign to get the final simplified expression:
±√((1 + cos(12x))/2)
So, the simplified expression using the half-angle formula is ±√((1 + cos(12x))/2).