prove tan(A/2) = (1 - cos(A))/sin(A)

Answer 1

To prove that tan(A/2) = (1 - cos(A))/sin(A), we'll use the half-angle identity for tangent.

The half-angle identity for tangent states that tan(A/2) = sin(A)/(1 + cos(A)).

To verify the given equation, we'll start with the half-angle identity for tangent and manipulate it into the desired form:

tan(A/2) = sin(A)/(1 + cos(A))

Let's multiply the numerator and denominator by (1 - cos(A)):

tan(A/2) = (sin(A)/(1 + cos(A))) * ((1 - cos(A))/(1 - cos(A)))

Simplifying the expression further:

tan(A/2) = sin(A)*(1 - cos(A))/[(1 + cos(A)) * (1 - cos(A))]

Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)):

tan(A/2) = sin(A)*(1 - cos(A))/[1 - cos^2(A)]

Using the Pythagorean identity (1 - cos^2(A) = sin^2(A)):

tan(A/2) = sin(A)*(1 - cos(A))/sin^2(A)

Canceling out the sin(A) term:

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

Thus, we have proved that tan(A/2) = (1 - cos(A))/sin(A).