Apply the definitions:
[tex] \cot(x) = \dfrac{\cos(x)}{\sin(x)} [/tex]
which implies
[tex] \cot\left(x-\dfrac{\pi}{2}\right) = \dfrac{\cos\left(x-\dfrac{\pi}{2}\right)}{\sin\left(x-\dfrac{\pi}{2}\right)} [/tex]
We also have the identities
[tex] \cos\left(x-\dfrac{\pi}{2}\right) = -\sin(x) [/tex]
[tex] \sin\left(x-\dfrac{\pi}{2}\right) = \cos(x) [/tex]
Which means
[tex] \cot\left(x-\dfrac{\pi}{2}\right) = \dfrac{\cos\left(x-\dfrac{\pi}{2}\right)}{\sin\left(x-\dfrac{\pi}{2}\right)} = \dfrac{-\sin(x)}{\cos(x)} = -\tan(x) [/tex]
