Question

Simplifique cos ²x. tg²x - cotg ²x. sen² x

Answer 1

Simplifique:

$cos^2(x)*tan^2(x)-ctg^2(x)*sin^2(x)$

Sabendo que:

$\begin{Bmatrix}sin^2(x)+cos^2(x)&=&1\\\\1+ctg^2(x)&=&\frac{1}{sin^2(x)}\\\\tan^2(x)+1&=&\frac{1}{cos^2(x)}\\\\cos(2x)&=&cos^2(x)-sin^2(x)\end{matrix}$

Substituindo, temos.

$cos^2(x)*\left(\frac{1}{cos^2(x)}-1\right)-\left(\frac{1}{sin^2(x)}-1\right)*sin^2(x)$

$1-cos^2(x)-1+sin^2(x)$

$sin^2(x)-cos^2(x)$

$-[cos^2(x)-sin^2(x)]$

Portanto a resposta é:

$\boxed{\boxed{cos^2(x)*tan^2(x)-ctg^2(x)*sin^2(x)=-cos(2x)}}$