Note: LHS = Left hand side, RHS = Right hand side

1/(1-sin x) + 1/(1+sin x) = 2sec^2 x

Consider LHS

1/(1-sin x) + 1/(1+sin x)

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

= 2/(1 - sin^2x)

= 2/cos^2x

= 2 sec^2x

=RHS

tan x + cot x = sec x csc x

Consider LHS

tan x + cot x

= sinx/cosx +cosx/sinx

= (sin^2x + cos^2x)/sinxcosx

= 1/sinxcosx

=(1/sinx)*(1/cosx)

= cscxsecx

= RHS

sec y + tan y = cos y/(1 - sin y)

Consider LHS

sec y + tan y

= 1/cosy + siny/cosy

= (cosy + cosysiny)/cos^2y

= (cosy + cosysiny)/(1 - sin^2y)

= [cosy(1 + siny)]/[(1 - siny)(1 + siny)].........the 1+siny cancels in the top and bottom

= cosy/(1 - siny)

= RHS