How do you prove ##(cosx/(1+sinx))+((1+sinx)/cosx)=2secx##?

Convert the left side into terms with common denominator and add (converting ##cos^2+sin^2## to ##1## along the way); simplify and refer to definition of ##sec = 1/cos##

##(cos(x)/(1+sin(x)))+((1+sin(x))/cos(x))##

##= (cos^2(x) + 1+2sin(x) +2sin(x) + sin^2(x))/(cos(x)( 1+sin(x)##

##= (2 +2sin(x))/(cos(x)(1+sin(x))##

##= 2/cos(x)##

##= 2* 1/cos(x)##

##= 2sec(x)##