Verified answer

1.

cos^2x(dy/dx)=y+3

dy/(y+3)=dx/(cos^2x)

dy/(y+3) =sec^2x dx

dy/(y+3) -sec^2x dx=0

ln|y+3| - tanx =k

ln (y+3) = tanx +k

y+3 = e^(tan x + k)

y = e^(tan x + k) - 3

2)

sinx/(1+y) * dy/dx=cosx

dy/(1+y) =cotx dx

dy/(1+y) -cotx dx=0

ln(1+y) -ln(sinx) = C

ln((1+y)/sinx)=C

(1+y)/sinx=e^C

1+y=e^C sinx

y=e^C sinx - 1

Sorry its late at night here, I'm tired and I'm not sure this one is right - just took its derivative and it didn't look like the differential eqn. Check it!!