1+sinx/cosx + cosx/ 1+sinx = 2secx

cosX / (1+ sinX)

multiply this term with (1-sinX) in Numerator and denominator

cosX *(1-sinX)/ (1+sinX )(1-sinX)

= cosX*(1-sinX) / (1-sin^2X)

= cosX*(1-sinx) / cos^2 X ...... acc. to identity sin^2X + cos^2X = 1 or 1 - sin^2X = cos^2X

now cosX canel out from numerator and denominator

= (1-sinX) / cosX

now see the problemm

(1 + sinX)/ cosX + cosX(1-sinX)

put the value of cosX(1-sinX) which is calculated above

(1+sinX) / cosX + (1-sinX)/cosX

Take LCM cosX

=( 1+sinX + 1- sinX)/ cosX

= 2 / cosX

and we know that 1/cosX = secX

= 2secX which is equal to Right Hand Side

so identity estaablished

This was very helpful, thank you!

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