To verify the identity we need the following identities:
i) [tex]\displaystyle{ \cot(x)= \frac{\cos x}{\sin x} [/tex]
ii) [tex]\displaystyle{ \sin (x-y)=\ sinx\cdot\ cosy -\ siny\cdot\ cosx[/tex]
iii) [tex]\displaystyle{ \cos (x-y)=\ cosx\cdot \ cosy +\ sinx\cdot\ siny[/tex].
Also, we have know that [tex]\displaystyle{ \sin \frac{ \pi }{2}=1 [/tex] and [tex]\displaystyle{ \cos \frac{ \pi }{2}=0.[/tex]
Thus, [tex]\displaystyle{ \cot(x-\frac{\pi}{2})= \frac{\cos (x-\frac{\pi}{2})}{\sin (x-\frac{\pi}{2})} [/tex]
By (ii) and (iii) we have:
[tex]\displaystyle{ \frac{\cos (x-\frac{\pi}{2})}{\sin (x-\frac{\pi}{2})}= \frac{\ cosx\cdot \ cos\frac{\pi}{2} +\ sinx\cdot\ sin\frac{\pi}{2}}{\ sinx\cdot\ cos\frac{\pi}{2} -\ sin\frac{\pi}{2}\cdot\ cosx} = \frac{\ sinx}{-\cos x} [/tex]
by simplifying [tex]\displaystyle{ \sin \frac{ \pi }{2}=1 [/tex] and [tex]\displaystyle{ \cos \frac{ \pi }{2}=0.[/tex]
Now, [tex]\displaystyle{ \frac{\ sinx}{-\cos x}[/tex] is clearly -tanx.
