Prove the following identity (1+2(sin a)(cos a))/(1-2(sin^2 a)=(cos a+sin a)/(cos a-sin a)

We recognize the formula of the cosine of double angle
for:

1-2(sin a)^2 = cos
2a

Also, we recognize the formula for the sine of double
angle:

2sina*cosa =
sin2a

We'll re-write the ratio from the left
side:

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

But cos 2a = (cos a)^2 - (sin
a)^2

cos 2a = (cos a - sin a)(cos a + sin
a)

We'll re-write the identity, substituting cos
2a:

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

We'll simplify and we'll
get:

(1 + sin 2a)/(cos a + sin a) = (cos a+sin
a)

1 + sin 2a = (cos a+sin
a)^2

We'll expand the
square:

1 + sin 2a = (cos a)^2+ 2sin a*cos a + (sin
a)^2

But (cos a)^2 + (sin a)^2 =
1

 1 + sin 2a = 1 + 2sin a*cos
a

 1 + sin 2a =  1 + sin 2a
q.e.d.