This is not an identity. Consider [tex]a=0^\circ[/tex], for which you get
[tex]\dfrac{\sin0^\circ+\sin(90-0)^\circ+1}{\sin0^\circ-\sin(90-0)^\circ}=\dfrac{0+1+1}{0-1}=-2[/tex]
but [tex]\cot0=\dfrac1{\tan0}=\dfrac10[/tex] is undefined.
