Question

Помогите пожалуйста упрастить выражение и пятое доказать ❤️❤️❤️❤️​

Answer 1

Ответ:

4) Упростим выражение:

$\frac{2*sin^{2}\alpha -1 }{sin^{2}\alpha-cos^{2}\alpha} =\frac{2*sin^{2}\alpha -(sin^{2}\alpha+cos^{2}\alpha) }{sin^{2}\alpha-cos^{2}\alpha} =\frac{2*sin^{2}\alpha -sin^{2}\alpha-cos^{2}\alpha}{sin^{2}\alpha-cos^{2}\alpha} =\frac{sin^{2}\alpha-cos^{2}\alpha}{sin^{2}\alpha-cos^{2}\alpha} =1$

5) Докажем тождество:

$(1+ctg^{2}\alpha +\frac{1}{cos^{2}\alpha})*sin^{2}\alpha*cos^{2}\alpha=(1+\frac{cos^{2}\alpha}{sin^{2}\alpha}+\frac{1}{cos^{2}\alpha})*sin^{2}\alpha*cos^{2}\alpha=\\=sin^{2}\alpha*cos^{2}\alpha+\frac{cos^{2}\alpha}{sin^{2}\alpha}*sin^{2}\alpha*cos^{2}\alpha+\frac{1}{cos^{2}\alpha}*sin^{2}\alpha*cos^{2}\alpha=\\=sin^{2}\alpha*cos^{2}\alpha+cos^{4}\alpha+sin^{2}\alpha=(1-cos^{2}\alpha)*cos^{2}\alpha+cos^{4}\alpha+sin^{2}\alpha=$

$=cos^{2}\alpha-cos^{4}\alpha+cos^{4}\alpha+sin^{2}\alpha=cos^{2}\alpha+sin^{2}\alpha=1$