$$CotA.CotB-1=-CotC(CotB+CotA)$$
$-\frac{1}{\left(\sin(A)\right)^{2}}$
$$\frac{\mathrm{d}}{\mathrm{d}A}(\frac{\cos(A)}{\sin(A)})$$
$$\frac{\sin(A)\frac{\mathrm{d}}{\mathrm{d}A}(\cos(A))-\cos(A)\frac{\mathrm{d}}{\mathrm{d}A}(\sin(A))}{\left(\sin(A)\right)^{2}}$$
$$\frac{\sin(A)\left(-\sin(A)\right)-\cos(A)\cos(A)}{\left(\sin(A)\right)^{2}}$$
$$-\frac{\left(\sin(A)\right)^{2}+\left(\cos(A)\right)^{2}}{\left(\sin(A)\right)^{2}}$$
$$-\frac{1}{\left(\sin(A)\right)^{2}}$$
$$-\left(\csc(A)\right)^{2}$$
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$\cot(A)$
