(tan2θ/tanθ−1) + (cot2θ/cotθ−1) = 1 + secθcosecθ.
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LHS:
(1+tan
2
θ)
2
tanθ
+
(1+cot
2
θ)
2
cotθ
We know that,
1+tan
2
θ=sec
2
θ
1+cot
2
θ=cosec
2
θ
(sec
2
θ)
2
tanθ
+
(cosec
2
θ)
2
cotθ
=
sec
4
θ
tanθ
+
cosec
4
θ
cotθ
=
cos
4
θ
1
tanθ
+
sin
4
θ
1
cotθ
=cos
4
θ⋅tanθ+cotθsin
4
θ
=cos
4
θ⋅
cosθ
sinθ
+
sinθ
cosθ
⋅sin
θ
=cos
3
θsinθ+cosθsin
3
θ
=cosθsinθ(cos
2
θ+sin
2
θ)
=cosθsinθ×1
∴sin
2
θ+cos
2
θ=1
=cosθsinθ.
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