Question

докажите тождество
$(1+\frac{1}{cos2x} +tg2x)(1-\frac{1}{cos2x} +tg2x)=2tg 2x$

Answer 1

Ответ:

$\Big(1+\dfrac{1}{cos2x}+tg2x\Big)\Big(1-\dfrac{1}{cos2x}+tg2x\Big)=\\\\\\=\Big((1+tg2x)+\dfrac{1}{cos2x}\Big)\Big((1+tg2x)-\dfrac{1}{cos2x}\Big)=(1+tg2x)^2-\Big(\dfrac{1}{cos2x}\Big)^2=\\\\\\=1+2tg2x+tg^22x-\dfrac{1}{cos^22x}=(\underbrace{1+tg^22x}_{\frac{1}{cos^22x}})+2tg2x-\dfrac{1}{cos^22x}=\\\\\\=\dfrac{1}{cos^22x}+2tg2x+\dfrac{2}{cos^22x}=2\, tg2x$