Question

cual es una expresión en términos de seno equivalente a
$\frac{ \tan( \alpha) - \cot( \alpha ) }{? \tan( \alpha ) + \cot( \alpha ) }$
a)
$2 { \sin}^{2} ( \alpha ) + 1$
b)
${ \sin }^{2} ( \alpha ) - 1$
c)
$2 { \sin}^{2} ( \alpha ) - 1$
d)
${ \sin}^{2} ( \alpha ) + 1$
​

Answer 1

Respuesta:

c)

Explicación paso a paso:

$\frac{\tan(\alpha)-\cot(\alpha)}{\tan(\alpha)+\cot(\alpha)}=\frac{\frac{\sin(\alpha)}{\cos(\alpha)})-\frac{\cos(\alpha)}{\sin(\alpha)}}{\frac{\sin(\alpha)}{\cos(\alpha)}+\frac{\cos(\alpha)}{\sin(\alpha)}}=\frac{\frac{\sin^{2}(\alpha)-\cos^{2} (\alpha)}{\cos(\alpha)\sin(\alpha)}}{\frac{\sin^{2}(\alpha)+\cos^{2}(\alpha)}{\cos(\alpha)\sin(\alpha)}}=\frac{\frac{\sin^{2}(\alpha)-\cos^{2} (\alpha)}{\cos(\alpha)\sin(\alpha)}}{\frac{1}{\cos(\alpha)\sin(\alpha)}}=\sin^{2}(\alpha)-\cos^{2}(\alpha)$

$=\sin^{2}(\alpha)-[1-\sin^{2}(\alpha)]=\sin^{2}(\alpha)-1+\sin^{2}(\alpha)=2\sin^{2}(\alpha)-1$