Question

Simplifique a expressão e encontre o valor de E:E = (sec x - cos x) . (cossec x - sen x) . (tg x + cotg x)
Fórmulas:sec ($\alpha$) = $\frac{1}{cos ( \alpha )}$cossec ($\alpha )$ = $\frac{1}{sen ( \alpha )}$cotg ($\alpha )$ = $\frac{1}{tg ( \alpha )}$

Answer 1

$E = (secx - cosx)(cossecx - senx)(tgx+cotgx)$
$E=([1/cosx]-cosx)([1/senx]-senx)(tgx+[1/tgx])$

$([1/cosx]-[cos^{2}x/cosx])([1/senx]-[sen^{2}x/senx])([tg^{2}x/tgx]+[1/tgx])$

$E=([1-cos^{2}x]/cosx)([1-sen^{2}x]/senx)([tg^{2}x+1]/tgx)$

Veja:

$sen^{2}x+cos^{2}x=1$
$sen^{2}x=1-cos^{2}x$
$cos^{2}x=1-sen^{2}x$

$tg^{2}x+1=sec^{2}x$

Substituindo:

$E=(sen^{2}x/cosx)(cos^{2}x/senx)(sec^{2}x/tgx)$

Simplificando:

$E=(senx/1)(cosx/1)([1/cos^{2}x]/tgx)$
$E=senx*cosx/(cos^{2}x*tgx)$
$E=senx*1/(cosx*senx/cosx)$
$E=senx/senx$
$E=1$