Question

(FGV-SP) A expressão "sec x - cos x/cossec x - sen x" é equivalente a: 

a) sec³ x. 
b) sen² x. 
c) tg³ x. 
d)1/tg x 
e)1/1 - tg² x.

Answer 1

$\frac{\sec\,x-\cos\,x}{cossec\,x-\sin\,x}=\\\\\frac{\frac{1}{\cos\,x}-\cos\,x}{\frac{1}{\sin\,x}-\sin\,x}=\\\\\frac{\frac{1-\cos^2x}{\cos\,x}}{\frac{1-\sin^2x}{\sin\,x}}=\\\\\frac{\frac{\sin^2x}{\cos\,x}}{\frac{\cos^2x}{\sin\,x}}=\\\\\frac{\sin^2x}{\cos\,x}\times\frac{\sin\,x}{\cos^2x}=\\\\\frac{\sin^3x}{\cos^3x}=\\\\\boxed{\tan^3x}$

A saber:

$\sin^2x+\cos^2x=1\Leftrightarrow\begin{cases}\sin^2x=1-\cos^2x\\\cos^2x=1-\sin^2x\end{cases}$