Prove that $1 + \frac{\cot^2 \alpha}{1 + \csc \alpha} = \csc \alpha$.

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Step 1: L.H.S $= 1 + \frac{\csc^2 \alpha - 1}{1 + \csc \alpha}$
Step 2: $= 1 + \frac{(\csc \alpha - 1)(\csc \alpha + 1)}{\csc \alpha + 1}$
Step 3: $= 1 + \csc \alpha - 1 = \csc \alpha = $ R.H.S

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