Question

Simplifique secx-cosx / cossecx-senx

Answer 1

$\mathsf{\dfrac{sec~x-cos~x}{cossec~x-sen~x}}\\\\\\\mathsf{\dfrac{\dfrac{1}{cos~x}-cos~x}{\dfrac{1}{sen~x}-sen~x}}\\\\\\\mathsf{\dfrac{\dfrac{1}{cos~x}-\dfrac{cos^2~x}{cos~x}}{\dfrac{1}{sen~x}-\dfrac{sen^2~x}{sen~x}}}\\\\\\\mathsf{\dfrac{\dfrac{1-cos^2~x}{cos~x}}{\dfrac{1-sen^2~x}{sen~x}}}\\\\\\\mathsf{\dfrac{1-cos^2~x}{cos~x}\cdot\dfrac{sen~x}{1-sen^2~x}}\\\\\\\mathsf{\dfrac{1-cos^2~x}{1-sen^2~x}\cdot\dfrac{sen~x}{cos~x}}\\\\\\\mathsf{\dfrac{1-cos^2~x}{1-sen^2~x}\cdot tg~x}$

Lembre da relação fundamental:

$\boxed{sen^2~x+cos^2~x=1}$

$\mathsf{\dfrac{sen^2~x}{cos^2~x}\cdot tg~x}\\\\\\\mathsf{tg^2~x\cdot tg~x}\\\\\\\boxed{\mathsf{tg^3~x}}$