$$cot^{3}A$$ is equal to

We know that $$cot 2A = \frac{cot^2 A - 1}{2 cot A}$$

and $$cot (A + B) = \frac{(cot A . cot B - 1)}{cot A + cot B}$$

Expression : $$cot 3A = cot (2A + A)$$

= $$\frac{(cot A . cot 2A - 1)}{cot A + cot 2A}$$

Now, $$(cot A . cot 2A - 1) = cot A (\frac{cot^2 A - 1}{2 cot A}) - 1$$

= $$\frac{cot^3 A - cot A - 2 cot A}{2 cot A} = \frac{cot ^3A - 3 cot A}{2 cot A}$$ --------------(i)

And, $$(cot A + cot 2A) = cot A + (\frac{cot^2 A - 1}{2 cot A})$$

= $$\frac{2 cot^2 A + cot^2 A - 1}{2 cot A} = \frac{3 cot^2 A - 1}{2 cot A}$$ ----------------(ii)

Dividing eqn(i) by (ii), we get :

= $$\frac{cot^3 A - 3 cot A}{3 cot^2 A - 1}$$

= $$\frac{3 cot A - cot^3 A}{1 - cot^2 A}$$