[tex]\sin q+\cos q\cot q=\sin q+\cos q\cdot\dfrac{\cos q}{\sin q}=\sin q+\dfrac{\cos^2q}{\sin q}\\\\=\dfrac{sin^2q}{\sin q}+\dfrac{\cos^2q}{\sin q}=\dfrac{\sin^2q+\cos^2q}{\sin q}=\dfrac{1}{\sin q}=\csc q\\\\Used:\\\cot q=\dfrac{\cos q}{\sin q}\\\\\sin^2q+\cos^2q=1\\\\\csc q=\dfrac{1}{\sin q}[/tex]
