- #1
Karol
- 1,380
- 22
Homework Statement
Prove:
$$\tan^{-1}(-x)=-\tan^{-1}(x)$$
Homework Equations
Inverse tangent: ##\tan(y)=x~\rightarrow~y=\tan^{-1}(x)##
The Attempt at a Solution
$$\tan(-x)=\frac{\sin(-x)}{\cos(-x)}=\frac{-\sin(x)}{\cos(x)}=-\tan(x)$$
I just change the unknown x to y:
$$\tan(-y)=\tan(y)$$
Now i have to translate it. we know:
$$\tan^{-1}(x)=\tan y~\rightarrow~-\tan^{-1}(x)=-\tan y$$
But:
$$\tan^{-1}(-x)~\rightarrow~\tan(y)=-x$$
Only from looking on the graph i can say ##\tan(-y)=-x## and finish but am i allowed to?