Finding the reciprocals using trigonometric functions and their inverses

Question

If $x\neq0$, how to find the reciprocal of $x$ only by using trigonometric functions and inverse trigonometric functions?

I have found only one answer, which is; $\tan (\cos ^{-1}(\sin(\tan^{-1}(x))))=\frac{1}{x}$.

Is there any other answer?

Answer 1

HINT

Recall that

• for $x>0$

$$\arctan x = \frac{\pi}2-\arctan \frac1x$$

• for $x<0$

$$\arctan x = -\frac{\pi}2-\arctan \frac1x$$

Answer 2

What about $\cot(\arctan x)$?