Question

A expressão sec^2 x - sen^2 x / cossec^2 x - cos^2 x é equivalente a:

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Answer 1

$\displaystyle \frac{\text{sec}^2\text x-\text{sen}^2\text x}{\text{cossec}^2\text x-\text{cos}^2\text x} \\\\\\ \frac{\displaystyle \frac{1}{\text{cos}^2\text x}-\text{sen}^2\text x}{\displaystyle \frac{1}{\text{sen}^2\text x}-\text{cos}^2\text x} \\\\\\ \frac{\displaystyle \frac{1-\text{sen}^2\text x.\text{cos}^2\text x}{\text{cos}^2\text x}}{\displaystyle \frac{1-\text{sen}^2\text x.\text{cos}^2\text x}{\text{sen}^2 \text x}}$

$\displaystyle \frac{[1-\text{sen}^2\text x.\text{cos}^2\text]}{\text{cos}^2\text x}.\frac{\text{sen}^2\text x}{[1-\text{sen}^2\text x.\text{cos}^2\text x]} \\\\\\\ \frac{\text{sen}^2\text x}{\text{cos}^2\text x} = \text{tg}^2\text x\\\\\\ \underline{\text{Portanto}}: \\\\ \frac{\text{sec}^2\text x-\text{sen}^2\text x}{\text{cossec}^2\text x-\text{cos}^2\text x}=\huge\boxed{\text{tg}^2\text x\ }\checkmark$

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