نکته: $\tan \left( a\pm \beta  \right)=\frac{\tan a\pm \tan \beta }{1\mp \tan a\tan \beta }$  نکته: $\tan 2a=\tan \left( a+a \right)=\frac{\tan a+\tan a}{1-\tan a\tan a}=\frac{2\tan a}{1-{{\tan }^{2}}a}$ نکته: $\cot \left( \frac{\pi }{2}+a \right)=-\tan a$  $\tan \left( \frac{\pi }{4}-x \right)=\frac{1}{3}\Rightarrow \frac{\tan \frac{\pi }{4}-\tan x}{1+\tan \frac{\pi }{4}\tan x}=\frac{1}{3}\Rightarrow \frac{1-\tan x}{1+\tan x}=\frac{1}{3}\Rightarrow 3-3\tan x=1+\tan x\Rightarrow \tan x=\frac{1}{2}$ $\cot \left( 2x+\frac{\pi }{2} \right)=-\tan 2x=-\frac{2\tan x}{1-{{\tan }^{2}}x}=-\frac{2\left( \frac{1}{2} \right)}{1-{{\left( \frac{1}{2} \right)}^{2}}}=-\frac{4}{3}$