This just simplifies to [tex]-tan(x)[/tex].
We are given
[tex]\dfrac{tan(-x)csc(-x) }{sec(-x)cot(-x)} [/tex].
Let's write each trig function in terms of sine and cosine.
[tex]\dfrac{ \frac{sin(-x)}{cos(-x)} \frac{1}{sin(-x)}}{ \frac{1}{cos(-x)} \frac{cos(-x)}{sin(-x)} } [/tex]
Cancel sin(-x) in the numerator and cos(-x) in the denominator.
[tex]\dfrac{\frac{1}{cos(-x)}}{\frac{1}{sin(-x)}}[/tex]
Simplify.
[tex]tan(-x)=-tan(x)[/tex]
