How can I prove this identity?

Question

(1 + sinØ + cosØ):(1 - sinØ + cosØ) =
(1 + sinØ) : cosØ

Answer 1

Work on the left hand side:
(1 + sinØ + cosØ)/(1 - sinØ + cosØ)
multiply both the numerator and denominator by (1-sinφ+cosφ) and simplify:
((1+cosφ)^2-sin&hi;^2)/(1-sinφ+cosφ)^2
=2cosφ(1+cosφ)/[2(1-sinφ)(1+cosφ)]
=2cosφ/[2(1-sinφ)]
=cosφ/(1-sinφ)
Now multiply top and bottom by (1+sinφ)
cosφ(1+sinφ)/(1-sin²φ)
=cosφ(1+sinφ)/cos²φ
=(1+sinφ)/cosφ