Find the general solution of the differential equations `e^xtanydx+(1-e^x)sec^2ydy=0`

Explanation

`e^(x)tanydx+(1-e^(x))sec^(2)ydy=0`
`implies e^(x)tany(dx=-(1-e^(x))sec^(2)ydy`
`implies (e^(x))/((e^(x)-1))dx=(sec^(2))/(tany)dy`
`implies int(e^(x))/((e^(x)-1))dx=int(sec^(2))/(tany)dy`
Let `e^(x)=1-timplies e^(x)dx=dt`
Let `tany=vimplies sec^(2)ydy=dv`
`:. int(1)/(t)dt=int(1)/(v)dv`
`implies log|t|=log|v|-log|C|`
`implies log|e^(x)-1|=log|tany|-log|C|`
`implies log|C(e^(x)-1)|=log|tany|`
`implies C(e^(x)-1)=tany`
which is the required general solution.