Question

Verificar la identidad:
tanT + 2cosTcscT = secTcscT + cotT

Answer 1

$\tan(x) + 2 \cos(x) \csc(x) = \sec(x) \csc(x) + \cot(x)$
$\tan(x) + 2 \cos(x) \csc(x) \\ = \frac{ \sin(x) }{ \cos(x) } + 2 \cos(x) \frac{1}{ \sin(x) } \\ = \frac{ \sin(x) }{ \cos(x) } + 2 \frac{ \cos(x) }{ \sin(x) } \\ = \frac{ \sin(x) \sin(x) + 2 \cos(x) \cos(x) }{ \cos(x) \sin(x) } \\ \frac{ { \sin(x) }^{2} + 2 { \cos(x) }^{2} }{ \cos(x) \sin(x) }$
$\frac{ { \sin(x) }^{2} + { \cos(x) }^{2} + { \cos(x) }^{2} }{ \cos(x) \sin(x) }$
ocupando que sen^2+cos^2=1
$\frac{1 + { \cos(x) }^{2} }{ \cos(x) \sin(x) } \\ = \frac{1}{ \cos(x) \sin(x) } + \frac{ { \cos(x) }^{2} }{ \cos(x) \sin(x) } \\ = \frac{1}{ \cos(x) } \frac{1}{ \sin(x) } + \frac{ \cos( x ) }{ \sin(x) } \\ = \sec(x) \csc(x) + \cot(x)$