Establish the identity
tanx/2=cscx-cotx
cscx - cotx = (1 - cosx)/sinx
That is the standard formula for tan x/2.
For a proof of that, see
http://oakroadsystems.com/twt/double.htm#TanHalf
To establish the identity tan(x/2) = csc(x) - cot(x), we can start by manipulating the right side of the equation until it matches the left side.
1. We will replace csc(x) and cot(x) with their respective definitions in terms of sin(x) and cos(x).
csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x).
Now, the equation becomes:
tan(x/2) = 1/sin(x) - cos(x)/sin(x).
2. To combine the two fractions on the right side, we need a common denominator, which is sin(x).
tan(x/2) = (1 - cos(x))/sin(x).
3. Next, we will express tan(x/2) in terms of sine and cosine using the half-angle identity for tangent.
tan(x/2) = (1 - cos(x))/sin(x) = sin(x/2)/cos(x/2).
4. Setting the two expressions equal to each other, we have:
sin(x/2)/cos(x/2) = (1 - cos(x))/sin(x).
5. Now, cross-multiply to simplify the equation:
sin^2(x/2) = (1 - cos(x))cos(x/2).
6. Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can rewrite sin^2(x/2) as 1 - cos^2(x/2).
1 - cos^2(x/2) = (1 - cos(x))cos(x/2).
7. Simplify the left side of the equation using the distributive property:
1 - cos^2(x/2) = cos(x/2) - cos(x)cos(x/2).
8. Rearrange the terms to one side of the equation:
cos^2(x/2) + cos(x)cos(x/2) - cos(x/2) + 1 = 0.
9. Factor the equation:
(cos(x/2) + 1)(cos(x/2) - 1) + cos(x/2)(cos(x) - 1) = 0.
10. Combine like terms in the second parentheses:
(cos(x/2) + 1)(cos(x/2) - 1) + cos(x/2)cos(x) - cos(x/2) = 0.
11. Simplify the terms:
cos^2(x/2) - 1 + cos(x/2)cos(x) - cos(x/2) = 0.
12. Combine like terms:
cos^2(x/2) + cos(x/2)cos(x) - cos(x/2) - 1 = 0.
13. Factor the equation further:
(cos(x/2) - 1)(cos(x/2) + cos(x) + 1) = 0.
Now, we have two possibilities:
1. cos(x/2) - 1 = 0, which implies cos(x/2) = 1.
2. cos(x/2) + cos(x) + 1 = 0.
Solving these equations will give us the possible values for x, which will establish the identity.