Question

Simplifique a expressão trigonométrica (sec (x)−cos( x)).(tg( x)+cotg( x)).sen( x) e
expresse-a unicamente em função de tg( x) .

Answer 1

Resposta:

Explicação passo-a-passo:

$(sec(x)-cos(x)).(tg(x)+cotg(x)).sen(x)\\\\rearranjando\\\\sen(x).(sec(x)-cos(x)).(tg(x)+cotg(x))\\\\\\onde\\sec(x)=\dfrac{1}{cos(x)}\\\\\\sen(x).(\dfrac{1}{cos(x)}-cos(x)).(tg(x)+cotg(x))\\\\\\(\dfrac{sen(x)}{cos(x)}-sen(x).cos(x)).(tg(x)+cotg(x))\\\\\\(tg(x)-sen(x).cos(x)).(tg(x)+cotg(x))\\\\\\tg^{2}(x)+tg(x).cotg(x)-tg(x).sen(x).cos(x)-sen(x).cos(x).cotg(x)\\\\\\onde\\\\tg(x).cotg(x)=\dfrac{sen(x)}{cos(x)}.\dfrac{cos(x)}{sen(x)}=1$

$tg^{2}(x)+1-\dfrac{sen(x)}{cos(x)} .sen(x).cos(x)-sen(x).cos(x).\dfrac{cos(x)}{sen(x)}\\\\\\tg^{2}(x)+1-sen^{2} (x)-cos^{2} (x)\\\\tg^{2}(x)+1-(sen^{2} (x)+cos^{2} (x))\\\\onde\\\\sen^{2} (x)+cos^{2} (x)=1\\\\tg^{2}(x)+1-(1)\\\\tg^{2}(x)\\\\Portanto\\\\\boxed{(sec (x)-cos( x)).(tg( x)+cotg( x)).sen(x)=tg^{2}(x)}$

Answer 2

$( \sec(x) - \cos(x) ) \: . \: ( \tan(x) + \cot(x) ) \: . \: \sin(x) \\ ( \frac{1}{ \cos(x) } - \cos(x) ) \: . \: ( \frac{ \sin(x) }{ \cos(x) } + \frac{ \cos(x) }{ \sin(x) } ) \: . \: \sin(x) \\ \frac{1 - \cos(x) {}^{2} }{ \cos(x) } \: . \: \frac{ \sin(x) {}^{2} + \cos(x) {}^{2} }{ \cos(x) \sin(x) } \: . \: \sin(x) \\ \frac{ \sin(x) {}^{2} }{ \cos(x) } \: . \: \frac{1}{ \cos(x) } \\ \frac{ \sin(x) {}^{2} }{ \cos(x) {}^{2} } \\ ( \frac{ \sin(x) }{ \cos(x) } ) {}^{2} \\ \boxed{\boxed{\boxed{ \tan(x) {}^{2} }}}$

atte. yrz