Question

Stiind ca x apartine (0;pi/4), sa se arate ca ctg 2x= (ctg x - tg x)/2. (Ex 253.)

Answer 1

[tex]ctg2x=\frac{ctgx-tgx}{2}\\ \\ ctg2x=\frac{\frac{cosx}{sinx}-\frac{sinx}{cosx}}{2}\\ \\ ctg2x=\frac{cos^2x-sin^2x}{2\sin x\cos x}\\ \\ ctg2x=\frac{cos2x}{sin2x}\\ ctg2x=ctg2x(A)[/tex]

Answer 2

$\displaystyle\frac{\cot x-\tan x}{2}=\frac{\frac{1}{\tan x}-\tan x}{2}=\frac{1-\tan^2x}{2\tan x}=\frac{1}{\tan 2x}=\cot 2x$