Prove the half-angle formula for tangent.

The half-angle formula for tangent is tan(x/2) = (1-cos(x))/sin(x).

To prove the half-angle formula for tangent, we start with the double-angle formula for tangent: tan(2x) = (2tan(x))/(1-tan^2(x)). Rearranging this formula, we get:

tan(x) = (tan(2x))/(2-2tan^2(x))

Next, we substitute x/2 for x in the above formula:

tan(x/2) = (tan(x))/(2-2tan^2(x/2))

We can simplify the denominator using the identity 1+tan^2(x/2) = sec^2(x/2):

tan(x/2) = (tan(x))/2(1-tan^2(x/2)) + (tan(x))/(2sec^2(x/2))

tan(x/2) = (tan(x))/2(1+tan(x/2))(1-tan(x/2)) + (sin(x/2))/(cos(x/2))

tan(x/2) = (sin(x))/(1+cos(x)) - (sin(x))/(1+cos(x))(1-tan(x/2))

tan(x/2) = (1-cos(x))/sin(x)

Therefore, we have proven the half-angle formula for tangent.