Resolver os seguintes exercícios a mão livre:
Simplifique a expressão
\frac{\cos ^{2}-\operatorname{cotg}(x)}{\operatorname{sen}^{2} x+\operatorname{tg}(x)}

Passo 1

O enunciado pede para simplificar a expressão trigonométrica
\frac{\cos^{2}(x)-\operatorname{cotg}(x)}{\operatorname{sen}^{2}(x)+\operatorname{tg}(x)}.
Para isso vamos usar as definições
\operatorname{cotg}(x)=\dfrac{\cos(x)}{\operatorname{sen}(x)},
\operatorname{tg}(x)=\dfrac{\operatorname{sen}(x)}{\cos(x)}
e a manipulação de frações.

Passo 2

Reescrevendo numerador e denominador em função de \operatorname{sen} e \cos:
Numerador:
\cos^{2}(x)-\operatorname{cotg}(x) =\cos^{2}(x)-\frac{\cos(x)}{\operatorname{sen}(x)}.
Denominador:
\operatorname{sen}^{2}(x)+\operatorname{tg}(x) =\operatorname{sen}^{2}(x)+\frac{\operatorname{sen}(x)}{\cos(x)}.

Passo 3

Colocamos cada um como fração única com denominador comum:
\cos^{2}(x)-\frac{\cos(x)}{\operatorname{sen}(x)} =\frac{\cos^{2}(x)\,\operatorname{sen}(x)-\cos(x)}{\operatorname{sen}(x)} =\frac{\cos(x)\bigl(\cos(x)\operatorname{sen}(x)-1\bigr)}{\operatorname{sen}(x)},
\operatorname{sen}^{2}(x)+\frac{\operatorname{sen}(x)}{\cos(x)} =\frac{\operatorname{sen}^{2}(x)\,\cos(x)+\operatorname{sen}(x)}{\cos(x)} =\frac{\operatorname{sen}(x)\bigl(\operatorname{sen}(x)\cos(x)+1\bigr)}{\cos(x)}.

Passo 4

Agora dividimos as duas frações:
\frac{\cos(x)\bigl(\cos(x)\operatorname{sen}(x)-1\bigr)/\operatorname{sen}(x)} {\operatorname{sen}(x)\bigl(\operatorname{sen}(x)\cos(x)+1\bigr)/\cos(x)} =\frac{\cos(x)\bigl(\cos(x)\operatorname{sen}(x)-1\bigr)}{\operatorname{sen}(x)} \cdot\frac{\cos(x)}{\operatorname{sen}(x)\bigl(\operatorname{sen}(x)\cos(x)+1\bigr)}.
Agrupando termos iguais:
=\frac{\cos^{2}(x)\,\bigl(\cos(x)\operatorname{sen}(x)-1\bigr)} {\operatorname{sen}^{2}(x)\,\bigl(\operatorname{sen}(x)\cos(x)+1\bigr)}.

Passo 5

Observe que \cos(x)\operatorname{sen}(x)-1=-(1-\cos(x)\operatorname{sen}(x)). Assim:
\frac{\cos^{2}(x)\,\bigl(\cos(x)\operatorname{sen}(x)-1\bigr)} {\operatorname{sen}^{2}(x)\,\bigl(\operatorname{sen}(x)\cos(x)+1\bigr)} =-\,\frac{\cos^{2}(x)\,\bigl(1-\operatorname{sen}(x)\cos(x)\bigr)} {\operatorname{sen}^{2}(x)\,\bigl(1+\operatorname{sen}(x)\cos(x)\bigr)}.
E \dfrac{\cos^{2}(x)}{\operatorname{sen}^{2}(x)}=\operatorname{cotg}^{2}(x). Portanto:

Passo 6

-\,\frac{\cos^{2}(x)}{\operatorname{sen}^{2}(x)} \;\frac{1-\operatorname{sen}(x)\cos(x)}{1+\operatorname{sen}(x)\cos(x)} =-\,\operatorname{cotg}^{2}(x)\,\frac{1-\operatorname{sen}(x)\cos(x)}{1+\operatorname{sen}(x)\cos(x)}.

Resposta

-\;\operatorname{cotg}^{2}(x)\;\frac{1-\operatorname{sen}(x)\cos(x)}{1+\operatorname{sen}(x)\cos(x)}.

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