Question

Let tan tanx=2 . Which is cot x?

Answer 1

1. Let $tantanx = 2$. Which is $cotx$?

Solution.

Firstly, let solve the equation

$$tantanx = 2$$

for $tanx$.

As $2 > 0$, then we ought to use the following roots of the equation $tany = a$:

$$y = arctana + \pi n, \quad n \in \mathbb{Z}.$$

So, one can receive:

$$tanx = arctan2 + \pi n, \quad n \in \mathbb{Z}.$$

Now, we shall use the trigonometric identity

$$tany \cdot coty = 1,$$

which observes for all $y \neq \frac{\pi k}{2}$, $k \in \mathbb{Z}$.

As $arctan2 + \pi n$ does not equal to $\frac{\pi k}{2}$ for any integer $n$ and $k$, then

$$cotx = \frac{1}{tanx} = \frac{1}{arctan2 + \pi n}, \quad n \in \mathbb{Z}.$$

Answer: $cotx = \frac{1}{arctan2 + \pi n}$, $n \in \mathbb{Z}$.