Question

cot(x+pi/2)=-tan x

Answer 1

\[\huge\color{blue}{Cot~~(x+ \frac{π}{2} ) =-Tan~x}\]
this?

Answer 2

\[\huge\color{blue}{ Cot~(x +90)=-Tan~x }\]
becuase π =180

Answer 3

\[Cot~(x+90)~~~~~->~~~~\frac{Cos(x+90) }{Sin(x+90)} \]
\[\sin(A+B)=\sin A \cos B + \cos A \sin B \]\[\cos(A+B)=\cos A \cos B - \sin A \sin B\]

\[\frac{Cos(x+90) }{Sin(x+90)}=\frac{\cos(x)\cos(90)-\sin(x)\sin(90) }{\sin(x)\cos(90)+\cos(x)\sin(90) }\]

Answer 4

\[\cos(90)=0~~~~and~~~\sin(90)=1.~~~~~~~~~So,\]the left side now becomes
\[\frac{-Sin(x) }{Cos(x)} ~~~~~or,~~~~~~~-\frac{Sin(x) }{Cos(x)}~~~~~->~~~~~-Tanx\]

Answer 5

So you now have
\[-Tan(x)=-Tan(x)\]
if you needed to verify the identity, verified.
if you needed to solve for x, then infinite number of solutions.