With figure,

img19.imageshack.us/img19/2135/81gd.png

prove:

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

To prove the equation tan(x/2) = sin(x)/(1+cos(x)), we can use the properties of trigonometric functions and the given figure image.

First, let's define the values in the figure:
- Consider the right triangle in the image. The two sides adjacent to the angle x/2 are labeled as a and b, while the hypotenuse (opposite to x/2) is labeled as c.
- Based on the figure, we can see that sin(x/2) = a/c and cos(x/2) = b/c.
- Now, consider the right triangle labeled x. The two sides adjacent to angle x are labeled as p and q, while the hypotenuse (opposite to x) is labeled as r.
- From this triangle, we can determine that sin(x) = q/r and cos(x) = p/r.

Now, let's work on proving the equation tan(x/2) = sin(x)/(1+cos(x)).

1. Start with the left-hand side (LHS):
tan(x/2) = sin(x/2) / cos(x/2)
a/c / b/c (from the definitions in the figure)

The c's cancel out:
a/b

2. Continue with the right-hand side (RHS):
sin(x)/(1+cos(x)) = q/r / (1 + p/r)

To simplify further, multiply both the numerator and denominator by r:
(q / r) / (r + p)

Substitute sin(x) = q/r and cos(x) = p/r:
(sin(x) / cos(x)) / (1 + cos(x))

Since tan(x) = sin(x) / cos(x), we have:
tan(x) / (1 + cos(x))

Therefore, the equation becomes:
tan(x/2) = tan(x) / (1 + cos(x))

From the above simplification, we can see that the LHS equals the RHS, which proves the equation tan(x/2) = sin(x)/(1+cos(x)).