Question

Why does `tan(x+(pi/2))=cot(x)`? or `cos((pi/2)-x)=sin(x)` and is it like that with `(3pi)/2` as well?

Please explain what's so special with `pi/2` and `(3pi)/2`

Answer 1

a. `tan (x + pi/2) = sin (x + pi/2)/(cos (x + pi/2))`
Reminder of trig identities:
`sin (x + pi/2) = cos x`
`cos (x + pi/2) = - sin x`
Therefor,
`tan (x + pi/2) = (cos x)/(-sin x) = - cot x`
To fully understand the property of complementary arcs, we prove theses below trig identities:
`cos (pi/2 + x) = cos (pi/2).cos x - sin (pi/2).sin x = - sin x`
`sin (pi/2 + x) = sin x.cos (pi/2) + cos x.sin (pi/2) = cos x`
`cos (pi/2 - x) = cos (pi/2.cos x + sin (pi/2).sin x = sin x`
`sin (pi/2 - x) = sin (pi/2).cos x + sin x.cos (pi/2) = cos x`