Verify the identity... cos(α - β) - cos(α + β) = 2 sin α sin β

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please show steps so I can see how to do this.

1)
cos(α-β) = cos(α)∙cos(β) + sin(α)∙sin(β)
and
cos(α+β) = cos(α)∙cos(β) - sin(α)∙sin(β)
so
cos(α-β) - cos(α+β) = (cos(α)∙cos(β) + sin(α)∙sin(β)) - (cos(α)∙cos(β) - sin(α)∙sin(β))
cos(α-β) - cos(α+β) = 2∙sin(α)∙sin(β)


2)
cot(θ) ∙ sec(θ) =

cos(θ)       1
---------- ∙ ---------- =
sin(θ)      cos(θ)

   1
-------- =
sin(θ)

csc θ

cos (a - b) = cos a cos b + sin a sin b
cos (a + b) = cos a cos b - sin a sin b

Subtracting, cos (a - b) - cos (a + b) = 2 sin a sin b