Prove each of the following identities:
cosec4 θ – cosec2 θ = cot4 θ + cot2 θ
Consider L.H.S. = cosec4 θ – cosec2 θ
= (cosec2 θ)2 – (cosec2 θ)
= (1 + cot2 θ)2 – (cosec2 θ)
= 1 + cot4 θ + 2cot2 θ – (cosec2 θ)
= 1 + cot4 θ + cot2 θ – (cosec2 θ – cot2 θ)
= 1 + cot4 θ + cot2 θ – 1
= cot4 θ + cot2 θ
= R.H.S.
Hence, proved.