Question

Упростить выражение
$\frac{1 - cos {}^{4} \alpha - sin {}^{4} \alpha }{ tg {}^{2} \alpha }$
​

Answer 1

Ответ:

$2cos^{4}\alpha$

Объяснение:

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

$=\frac{sin^{2}\alpha}{tg^{2}\alpha}+\frac{sin^{2}\alpha*cos^{2}\alpha}{tg^{2}\alpha}-\frac{sin^{4}\alpha}{tg^{2}\alpha}=\frac{sin^{2}\alpha*cos^{2}\alpha}{sin^{2}\alpha}+\frac{sin^{2}\alpha*cos^{2}\alpha*cos^{2}\alpha}{sin^{2}\alpha}-\frac{sin^{2}\alpha*sin^{2}\alpha*cos^{2}\alpha}{sin^{2}\alpha}=$

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

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