Question

Olá gente podem me ajudar?
$\frac{\frac{\sin ^2\left(x\right)}{\cos \left(x\right)}}{\frac{\cos \left(x\right)+1}{\cos \left(x\right)}}$

Answer 1

Olá,

$\mathrm{Usar\:a\:seguinte\:identidade}:\quad \frac{\sin \left(x\right)}{\cos \left(x\right)}=\tan \left(x\right)$
$=\frac{\sin \left(x\right)\tan \left(x\right)}{\frac{\cos \left(x\right)+1}{\cos \left(x\right)}}$

$\mathrm{Aplicar\:as\:propriedades\:das\:fracoes}:\quad \frac{a}{\frac{b}{c}}=\frac{a\cdot \:c}{b}$
$=\frac{\sin \left(x\right)\tan \left(x\right)\cos \left(x\right)}{\cos \left(x\right)+1}$

$\mathrm{Usar\:a\:seguinte\:identidade}:\quad \cos \left(x\right)\sin \left(x\right)=\frac{\sin \left(2x\right)}{2}$
$=\frac{\frac{\sin \left(2x\right)}{2}\tan \left(x\right)}{1+\cos \left(x\right)}$

$\mathrm{Multiplicar\:}\frac{\sin \left(2x\right)}{2}\tan \left(x\right)\::\quad \frac{\sin \left(2x\right)\tan \left(x\right)}{2}$
$\frac{\sin \left(2x\right)}{2}\tan \left(x\right)$


$\mathrm{Multiplicar\:fracoes}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}$
$=\frac{\sin \left(2x\right)\tan \left(x\right)}{2}$


$=\frac{\frac{\sin \left(2x\right)\tan \left(x\right)}{2}}{1+\cos \left(x\right)}$


$\mathrm{Aplicar\:as\:propriedades\:das\:fracoes}:\quad \frac{\frac{b}{c}}{a}=\frac{b}{c\:\cdot \:a}$
Que é igual a:
$=\frac{\sin \left(2x\right)\tan \left(x\right)}{2\left(\cos \left(x\right)+1\right)}$

Answer 2

$\displaystyle\frac{\frac{\sin^2{x}}{\cos{x}}}{\frac{\cos{x}+1}{\cos{x}}}=\dfrac{\sin^2{x}}{\cos{x}+1}=\dfrac{1-\cos^2{x}}{\cos{x}+1}=\dfrac{(1+\cos{x})(1-\cos{x})}{(\cos{x}+1)}=1-\cos{x}$