Final answer:
To prove cot(x/2) - 2cot(x) = tan(x/2), we use the reciprocal and double angle formulas for tangent and simplify to obtain tan(x/2), confirming the identity.
Step-by-step explanation:
To prove the identity cot(x/2) - 2cot(x) = tan(x/2), we can use various trigonometric identities. First, we convert cotangent to its reciprocal function and use the double angle formula for tangent. The key identities are:
- cot(θ) = 1/tan(θ)
- tan(2θ) = 2tan(θ) / (1 - tan2(θ))
- tan(θ/2) = (1 - cos(θ)) / sin(θ)(half-angle identity for tangent)
We start by expressing the cotangent functions in terms of tangent:
cot(x/2) - 2cot(x) = 1/tan(x/2) - 2(1/tan(x))
We then represent tan(x) as tan(2 * (x/2)) and apply the double angle formula:
1/tan(x/2) - (1 - tan2(x/2)) / (2tan(x/2))
After finding a common denominator and simplifying, we get:
tan(x/2), which completes the proof.