Question

1+sin a- cos a/1+sin a +cos a eual to ?.
How?

Answer 1

Multiply both numerator and denominator by the conjugate, (1+sin(a)-cos(a))

To reduce typing, will use s=sin(a), c=cos(a),
so
(1+sin(a)- cos(a))/(1+sin(a)+cos(a))
=(1+s-c)/(1+s+c)
=(1+s-c)²/((1+s+c)(1+s-c))
=(1+s²+2s+c²-2c(1+s))/((1+s)²-c²)
now use s²+c²=1
=((2+2s)-2c(1+s))/(1+2s+s²-c²)
=2(1-c)(1+s)/(2s(1+s))
cancel 2(1+s) assume s≠-1
=(1-c)/s
=(1-cos(a))/sin(a)

Check my work.