Prove that tan3 θ/1 + tan2 θ + cot3 θ/1 + cot2θ = sec θ. cosec θ – 2 sin θ.cos θ
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Step-by-step explanation:
L.H.S. = tan3 θ/1 + tan2 θ + cot3 θ/1 + cot2 θ
= tan3 θ/sec2 θ + cot3 θ/cosec2 θ
= sin3 θ/cos3 θ × cos2 θ + cos3 θ/sin3 θ × sin2 θ
= sin3 θ/cos θ + cos3 θ/sin θ = sin4 θ + cos4 θ/sin θ.cos θ
= [sin2 θ + cos2 θ]2 – 2 sin2 θ. cos2 θ/sin θ.cos θ
= [(1)2 – 2 sin2 θ.cos2 θ]/sin θ.cos θ
= 1 – 2 sin2 θ.cos2 θ/sin θ.cos θ
= (1 – 2 sin2 θ.cos2 θ)/sin θ.cos θ
= 1/sin θ.cos θ = 2 sin2 θ.cos2 θ/sin θ.cos θsec
=θ.cosec θ – 2 sin θ.cos θ
= R.H.S.
Hence proved.
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