I'm going to assume you meant sin(x+y)/(sinxcosy)=1+cotxtanysin(x+y)sinxcosy=1+cotxtany and just forgot parentheses on the left side denominator and the plus on the right side, otherwise the identity is false.
Let's start working the left hand side by using the sine addition formula sin(alpha+beta)=sinalphacosbeta+cosalphasinbetasin(α+β)=sinαcosβ+cosαsinβ
sin(x+y)/(sinxcosy)=(sinxcosy+cosxsiny)/(sinxcosy)sin(x+y)sinxcosy=sinxcosy+cosxsinysinxcosy
Then split the numerator over the denominator
(sinxcosy+cosxsiny)/(sinxcosy)=(sinxcosy)/(sinxcosy)+(cosxsiny)/(sinxcosy)sinxcosy+cosxsinysinxcosy=sinxcosysinxcosy+cosxsinysinxcosy
Simplify and factor
(sinxcosy)/(sinxcosy)+(cosxsiny)/(sinxcosy)=1+(cosx/sinx)(siny/cosy)sinxcosysinxcosy+cosxsinysinxcosy=1+(cosxsinx)(sinycosy)
And knowing that tantheta=sintheta/costhetatanθ=sinθcosθ and cottheta=costheta/sinthetacotθ=cosθsinθ,
1+(cosx/sinx)(siny/cosy)=1+cotxtany1+(cosxsinx)(sinycosy)=1+cotxtany
Therefore the left side is equal to the right side. Q.E.D.