Question

Упростите выражение (sin2a+tg2a)/(1+cos2a)

Answer 1

$\dfrac{\sin2a+\mathrm{tg}2a}{1+\cos2a} =\dfrac{\sin2a+\frac{\sin2a}{\cos2a} }{1+2\cos2a} =\dfrac{\sin2a\cdot (1+\frac{1}{\cos2a})}{1+\cos2a} = \\\ =\dfrac{\sin2a\cdot \frac{\cos2a+1}{\cos2a}}{1+\cos2a} = \dfrac{\sin2a\cdot (\cos2a+1)}{\cos2a\cdot (1+\cos2a)} =\dfrac{\sin2a}{\cos2a} =\mathrm{tg}2a$

Answer 2

Числитель и знаменатель умножим на ctg2α:

$\frac{ctg2\alpha(sin2\alpha+tg2\alpha)}{ctg2\alpha(1+cos2\alpha)} =\frac{cos2\alpha+1}{ctg2\alpha(1+cos2\alpha)} =\frac{1}{ctg2\alpha} =tg2\alpha$