Prove the following identity: (1+ sin x + cos x) / (1+ sin x - cos x) = cot x/2

We know that 1 + cos x = 2(cos x/2)^2
(1)

We'll write sin x = sin 2(x/2) = 2sin(x/2)*cos (x/2)
(2)

We'll add (1) + (2):

2(cos
x/2)^2 + 2sin(x/2)*cos (x/2)

We'll factorize by 2cos
(x/2):

1+ sin x + cos x = 2cos (x/2)[cos (x/2) +
sin(x/2)]

We'll write the terms from denominator
as:

1 - cos x = 2(sin x/2)^2
(3)

sin x = sin 2(x/2) = 2sin(x/2)*cos
(x/2)

We'll add (3) +
(2):

2(sin x/2)^2 + 2sin(x/2)*cos
(x/2)

We'll factorize by 2sin
(x/2):

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

We'll re-write the numerator and denominator of the
ratio:

2cos (x/2)[cos (x/2) + sin(x/2)]/2sin(x/2)*[sin(x/2)
+ cos (x/2)]

We'll simplify the
brackets:

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

We'll simplify and we notice that the ratio
cos (x/2)/sin(x/2) = cot (x/2)

(1+ sin x +
cos x) / (1+ sin x - cos x) = cot (x/2) q.e.d.