Cos (α-90) = cos [α-(π/2)] This step is because π = 180 in trigonometric function.

Then we add a minus sign to α-(π/2) to make it -[-α+(π/2)].

Then α-(π/2)= -[-α+(π/2)]

So cos [α-(π/2)]=cos{ -[-α+(π/2)]}

Then we regard -α+(π/2) as a whole and make it = t.

So cos{ -[-α+(π/2)]}=cos(-T) According to the inductive formula, we can get cos(-α)=cosα.

So cos(-T)=cosT= cos[-α+(π/2)] According to the inductive formula cos[(π/2)-α]=sinα.

So cos[-α+(π/2)] =sinα.

So cos [α-(π/2)] = cos {-[-α+(π/2)]} = cos [-α+(π/2)] = sin α.

I hope you can adopt it. Thank you.