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LammettHash
One way to do it:

[tex]\cot\left(x-\dfrac\pi2\right)=\dfrac{\cos\left(x-\frac\pi2\right)}{\sin\left(x-\frac\pi2\right)}[/tex]
[tex]=\dfrac{\cos x\cos\frac\pi2+\sin x\sin\frac\pi2}{\sin x\cos\frac\pi2-\cos x\sin\frac\pi2}[/tex]
[tex]=\dfrac{\sin x}{-\cos x}[/tex]
[tex]=-\tan x[/tex]

Answer:

By definition, cot = cos/sin. Next, use the angle sum identities: cos(x +/- y) = cosx cosy -/+ sinx sin y; sin(x +/- y) = sinx cosy +/- cosx siny. Simplify where applicable, namely sin(pi/2) = 1 and cos(pi/2) = 0. This leaves you with sin/-cos, or -sin/cos. By definition, tan = sin/cos, so we're done.


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