Quick trig identities help

I have just done a worksheet full of them but I have still not figured out how to do these 2 ID's Can you guys help guide me through this?

tan(x) + cos(x)/(1+sin(x)=sec(x)

cot(x)/(1-tan(x)) + tan(x)/(1-cot(x) = 1 + tan(x) + cot(x)

Thanks, I've spend an hour trying to figure these two and im so dizzy.

tan(x) + cos(x)/(1+sin(x)=

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

(sinx(1+sinx)+cos^2x) / (cosx(1+sinx))=

(sinx+sin^2x+cos^2x) / (cosx(1+sinx))=

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

1/cosx=secx

cot(x)/(1-tan(x)) + tan(x)/(1-cot(x))=

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

(cos^2x) / ((sinx(cosx-sinx)) + (sin^2x) / ((cosx(sinx-cosx)) =

- cos^2xcosx / (sinxcosx(sinx-cosx)) + sin^2xsinx / ((sinxcosx(sinx-cosx)) =

(sin^3x-cos^3x) / (sinxcosx(sinx-cosx)) =

((sinx-cosx)(sin^2x+sinxcosx+cos^2x)) / (sinxcosx(sinx-cosx)) =

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

(1+sinxcosx) / sinxcosx =

1+(1/sinxcosx)=

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

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

1+sinx/cosx+cosx/sinx=

1+tanx+cotx