\(\frac{sinb-cosa.sin\left(a+b\right)}{cosb-cosa.cos\left(a+b\right)}=\frac{sinb-cosa\left(sina.cosb+cosa.sinb\right)}{cosb-cosa.\left(cosa.cosb-sina.sinb\right)}\)
\(=\frac{sinb-cos^2a.sinb-sina.cosa.cosb}{cosb-cos^2a.cosb+sina.cosa.sinb}=\frac{sinb\left(1-cos^2a\right)-sina.cosa.cosb}{cosb\left(1-cos^2a\right)+sina.cosa.sinb}\)
\(=\frac{sinb.sin^2a-sina.cosa.cosb}{cosb.sin^2a+sina.cosa.sinb}=\frac{-sina\left(cosa.cosb-sina.sinb\right)}{sina\left(sina.cosb+cosa.sinb\right)}\)
\(=\frac{-cos\left(a+b\right)}{sin\left(a+b\right)}=-cot\left(a+b\right)\)