Question

How do you verify `tan (x+pi/2)= -cot x`?

Answer 1

By applying the basic trigonometric relations. see below

Explanation:

head

Key-relation 1. `tanx=sinx/cosx`

Key-relation 2. `cotx=1/tanx=cosx/sinx`

Key-relation 3. `cos(a+b)=cosa*cosb -sina*sinb`

Key-relation 4. `sin(a+b)=cosa*sinb + sina*cosb`

Import results

Important result 1. `cos (pi/2)=0`
Important result 1. `sin (pi/2)=1`

Gathering

By using all the knowledge left so far and some mathematical tricks, we have:

`sin(x+(pi/2))=cosx*sin(pi/2) + sinx*cos(pi/2)`
`sin(x+(pi/2))=cosx`

Further:

`cos(x+(pi/2))=cosx*cos(pi/2) - sinx*sin(pi/2)`

`cos(x+(pi/2))=- sinx`

Finally:

`tan(x+(pi/2))=sin(x+(pi/2))/cos(x+(pi/2))`

`tan(x+(pi/2))=-cosx/sinx=cotx`

End of the proof!