Question

Verify the identity cot(x-pi/2)=-tan x

Answer 1

To verify the identity we need the following identities:

i)
`\displaystyle{ \cot(x)= (\cos x)/(\sin x)`

ii)
`\displaystyle{ \sin (x-y)=\ sinx\cdot\ cosy -\ siny\cdot\ cosx`

iii)
`\displaystyle{ \cos (x-y)=\ cosx\cdot \ cosy +\ sinx\cdot\ siny`.

Also, we have know that
`\displaystyle{ \sin ( \pi )/(2)=1` and
`\displaystyle{ \cos ( \pi )/(2)=0.`


Thus,
`\displaystyle{ \cot(x-(\pi)/(2))= (\cos (x-(\pi)/(2)))/(\sin (x-(\pi)/(2)))`

By (ii) and (iii) we have:


`\displaystyle{ (\cos (x-(\pi)/(2)))/(\sin (x-(\pi)/(2)))= (\ cosx\cdot \ cos(\pi)/(2) +\ sinx\cdot\ sin(\pi)/(2))/(\ sinx\cdot\ cos(\pi)/(2) -\ sin(\pi)/(2)\cdot\ cosx) = (\ sinx)/(-\cos x)`

by simplifying
`\displaystyle{ \sin ( \pi )/(2)=1` and
`\displaystyle{ \cos ( \pi )/(2)=0.`

Now,
`\displaystyle{ (\ sinx)/(-\cos x)` is clearly -tanx.