Question

Доказать тождество (1+cos2t-sin2t)/(1+sin2t+cos2t)=tg(п/4-t)

Answer 1

$\frac{1 + cos2x - sin2x}{1 + cos2x + sin2x} = \frac{1 + cos^{2}x - sin^{2}x - 2 sinxcosx}{1 + cos^{2}x - sin^{2}x + 2 sinxcosx} = \frac{2cos^{2}x - 2sinxcosx}{2cos^{2}x + 2sinxcosx} =$

$=\frac{2cosx (cosx - sinx)}{2cosx (cosx + sinx)} = \frac{cosx - sinx}{cosx + sinx} = \frac{1-tgx}{1+tgx} = \frac{tg\frac{\pi}{4} - tgx}{1 + tg\frac{\pi}{4}tgx} = tg(\frac{\pi}{4} - x)$