(1+sinx)/(1-sinx) = [(1+sinx)(1+sinx)]/[((1-sinx)(1+sinx)] (Multiplying 1+sinx in the denominator and numerator)
=[(1+sinx)^2]/(1-sin^2(x)) (Using the identity (a-b)(a+b)=a^2-b^2)
=[(1+sinx)^2]/cos^2(x) (Using the trig identity cos^2(x) = 1 - sin^2(x))
= [(1+sinx)/cosx]^2 (Using a^2/b^2 = (a/b)^2)
Hence,
sqrt((1+sinx)/(1-sinx)) = sqrt([(1+sinx) /cos(x)]^2)
= |(1+sinx)/cosx| (Using sqrt(a^2)=|a|)
=|1+sinx|/|cosx| (Using |a/b| = |a|/|b|)
=(1+sinx)/|cosx| (|1+sinx|=1+sinx since 1+sinx >=0 as sin(x)>=-1)
Now we can conclude that
sqrt((1+sinx)/(1-sinx))=(1+sinx)/|cosx|
