Question

How do you simplify `tanx/(1-tanx)+cosx/(sinx-cosx)`?

Answer 1

`tan(x)/(1-tan(x))+cos(x)/(sin(x)-cos(x))=-1` for `x !in {pi/2+pik, pi/4+pik}`

Explanation:

For `x !in {pi/2+pik, pi/4+pik}`

`tan(x)/(1-tan(x))+cos(x)/(sin(x)-cos(x))`

`=(tan(x)(-cos(x)))/((1-tan(x))(-cos(x)))+cos(x)/(sin(x)-cos(x))`

`=-sin(x)/(sin(x)-cos(x))+cos(x)/(sin(x)-cos(x))`

`=(cos(x)-sin(x))/(sin(x)-cos(x))`

`=-(sin(x)-cos(x))/(sin(x)-cos(x))`

`=-1`