First rewrite the right side, replacing cotθ with cosθ/sinθ to get:
This simplifes to 1-2sinθcosθ, do you see that?
sin2θ+cos2θ =1 so you can replace the "1' above to get: sin2θ+cos2θ-2sinθcosθ
Rearrange and you have sin2θ-2sinθcosθ+cos2θ
which factors to (sinθ-cosθ)2 and you are done!
Here's the same problem but approaching from the left side, I think it is easier.
First off you will want to FOIL the left side to get: sin2θ-2sinθcosθ+cos2θ
By re-grouping like this: [sin2θ+cos2θ]-2sinθcosθ it is easy to see that this simplifies to: 1-2sinθcosθ
Now look at the right side:
since cotθ=cosθ/sinθ, 1-2sin2θ*cotθ can be re-written as:
which matches what we had for the left side above. This way does not require the double angle rule