Explanation:

Whenever the target expression involves large powers of sinθ or cosθ, we can be sure that from the input expression we would get either the value of θ or the value of sinθ, as otherwise such large powers can't be evaluated.
With this prior knowledge we proceed to expand the given expression,
sinθ+cosecθ=2,
Or, sin²θ−2sinθ+1=(sinθ−1)²=0
Or, sinθ=1, and so, cosecθ=1.
Putting these convenient values in the target expression we get,
E=1+1=2