Question

simplifique a equação (1/cos² x * cossec² x) - sec² x +2

Answer 1

Olá

$\displaystyle \mathsf{\left(\frac{1}{cos^2x\cdot csc^2x} \right)-sec^2x+2}\\\\\\\text{Reescrevendo a cossecante como}\\\\\mathsf{csc^2x= \frac{1}{sen^2x} }\\\\\text{Substituindo}\\\\\\\mathsf{\left(\frac{1}{cos^2x\cdot \frac{1}{sen^2x} } \right)-sec^2x+2}\\\\\\\mathsf{\left(\frac{1}{ \frac{cos^2x}{sen^2x} } \right)-sec^2x+2}$

Divisão de fração, multiplica a primeira pelo inverso da segunda

$\displaystyle \mathsf{\left(\frac{sen^2x}{cos^2x} \right)-sec^2x+2}\\\\\\ \mathsf{\frac{sen^2x}{cos^2x} -sec^2x+2}\\\\\\\text{Reescrevendo a secante como}\\\\\mathsf{sec^2x= \frac{1}{cos^2x} }\\\\\\\mathsf{\frac{sen^2x}{cos^2x} - \frac{1}{cos^2x} +2}\\\\\\\text{Faz o MMC}\\\\\\\mathsf{\frac{sen^2x-1}{cos^2x} +2}$

Identidade trigonométrica fundamental

$\mathsf{sen^2x+cos^2x=1}\\\\\mathsf{sen^2x=1-cos^2x}\\\\\mathsf{sen^2x-1=-cos^2x}$

Substituindo

$\displaystyle \mathsf{\frac{sen^2x-1}{cos^2x} +2}\\\\\\\mathsf{\frac{-cos^2x}{cos^2x} +2}\\\\\\\text{Simplifica}\\\\\\\mathsf{\underbrace{ \frac{-cos^2x}{cos^2x} }_{=-1}~+~2}\\\\\\\mathsf{-1+2}\\\\\boxed{\mathsf{1}}\\\\\\\text{Ou, como vimos acima:}\\\\\boxed{\mathsf{1=cos^2x+sen^2x}}$