What is $\tan^{3} (\arctan{x})$? This is part of the answer to an integral I am working out, and I was wondering if it would simplify so I can make the answer look nicer. So far I am wondering if will simplify to $\tan^{2}(x)$ because I know that $\arctan(\tan{x})$ is just $x$.
What is $\tan^{3} (\arctan{x})$?
-5
$\begingroup$
trigonometry
3 Answers
0
It really depends on what the superscript 3 denotes. Unfortunately with the trigonometric functions a superscript is often placed in that position to indicate numerical powers of the function's output (when really such an exponent should be placed after the close-bracket). In broader contexts, such a superscript would denote repeated application of the function (or its inverse if negative). In this case, it is precisely as you suggest. $$\tan^3 \arctan x = \tan \tan \tan \arctan x = \tan \tan x = \tan^2 x$$
2
$$\tan^3(\arctan x)=(\tan(\arctan x))^3=x^3$$
$$\tan^3(\arctan x)\ne\tan^2(x)\tan(\arctan x)$$
1
It holds that $$\tan^3(\arctan x)= (\tan(\arctan x))^3= x^3$$