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Answer 1 · 2017-02-08T23:04:06

See below

Cross multiply:

#color(green)(cosx)/(color(red)(1+sinx)) = (color(red)(1-sinx))/color(green)(cosx)#

#implies color(green)(cos^2 x) = color(red)((1 + sin x)(1 - sin x))#

#implies color(green)(cos^2 x) = color(red)(1 - sin^2 x)#

And that is the Pythagorean Identity :)

Answer 2 · 2017-02-09T03:50:22

#LHS=cosx/(1+sinx)#

#=(cosx(1-sinx))/((1+sinx)(1-sinx))#

#=(cosx(1-sinx))/(1-sin^2x)#

#=(cosx(1-sinx))/cos^2x#

#=(1-sinx)/cosx=RHS#

proved