Prove that cot beta = 2 tan (alpha-beta) if 2tan beta + cot beta = 1.
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2tan(α−β)
=2[
tanα−tanβ
1+tanα.tanβ
]
=2[
2tanβ+cotβ−tanβ
1+(2tanβ+cotβ).tanβ
]
[as,tanα=2tanβ+cotβ]
=2[
tanβ+cotβ
1+2
tan
2
β+1
]
=
2(tanβ+cotβ)
2+2
tan
2
β
=
2(tanβ+
1
tanβ
)
2(1+
tan
2
β)
=
1
tanβ
=cotβ
So,
cotβ=2tan(α−β)
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