Question

упростить выражение (sin(pi/4-a)+cos(pi/4-a))/(sin(pi/4-a)-cos(pi/4-a))

Answer 1

$<var>\frac{\sin(\frac{\pi}{4}-a)+\cos(\frac{\pi}{4}-a)}{\sin(\frac{\pi}{4}-a)-\cos(\frac{\pi}{4}-a)}=\\ =\frac{(\sin(\frac{\pi}{4}-a)+\cos(\frac{\pi}{4}-a))^2}{(\sin(\frac{\pi}{4}-a)-\cos(\frac{\pi}{4}-a))(\sin(\frac{\pi}{4}-a)+\cos(\frac{\pi}{4}-a))}=\\ =\frac{\sin^2(\frac{\pi}{4}-a)+\cos^2(\frac{\pi}{4}-a)+2\sin(\frac{\pi}{4}-a)\cos(\frac{\pi}{4}-a)}{\sin^2(\frac{\pi}{4}-a)-\cos^2(\frac{\pi}{4}-a)}=\\ =\frac{1+\sin(\frac{\pi}{2}-2a)}{-\cos(\frac{\pi}{2}-2a)}=-\frac{1+\cos 2a}{\sin 2a} </var>$

$<var>-\frac{1+\cos 2a}{\sin 2a} = -\frac{cos^2 a + \sin^2 a + \cos^2 a - \sin^2 a}{2\sin a \cos a}=\\ =-\frac{2\cos^2a}{2\sin a \cos a}=-\frac{\cos a}{\sin a} = -\cot a</var>$

Answer 2

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 ответ -ctg a