Explain $\tan^2(\tan^{-1}(x))$ becoming $x^2$

Question

How does $\tan^2(\tan^{-1}(x))$ become $x^2$?

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I feel that the answer should contain a tan somewhere and not just simply $x^2$. "Why?" you might ask, well I thought that $\tan^2(\theta)$ was a special function that has to be rewritten a specific way.

Answer 1

By definition $\tan(\tan^{-1}(x)) = x$ because $\tan^{-1}(x)$ does not mean $1/\tan(x)$ but the inverse function to $\tan(x)$.

Another convention is that $(\tan(x))^2$ is too long for some people and they will just write $\tan^2(x)$.

Putting these two things together, you get

$$\tan^2(\tan^{-1}(x)) = (\tan(\tan^{-1}(x)))^2 = (x)^2 = x^2$$

Answer 2

Because $\tan^{-1}x $ is not the reciprocal of $\tan x$, but the inverse function $\arctan x$.