Matemática
Leticia798
7

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

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paula122

Simplifique: [latex]cos^2(x)*tan^2(x)-ctg^2(x)*sin^2(x)[/latex] Sabendo que: [latex]\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}[/latex] Substituindo, temos. [latex]cos^2(x)*\left(\frac{1}{cos^2(x)}-1\right)-\left(\frac{1}{sin^2(x)}-1\right)*sin^2(x)[/latex] [latex]1-cos^2(x)-1+sin^2(x)[/latex] [latex]sin^2(x)-cos^2(x)[/latex] [latex]-[cos^2(x)-sin^2(x)][/latex] Portanto a resposta é: [latex]\boxed{\boxed{cos^2(x)*tan^2(x)-ctg^2(x)*sin^2(x)=-cos(2x)}}[/latex]

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