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Question

If α+β=γ and tanγ=22, a is the arithmetic and b is the geometric mean respectively between tanα and tanβ, then the value of a3(1b2)3 is equal to

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Solution

Given tanα,b,tanβ are in GP
b2=tanαtanβ ....(1)
Also given tanα,a,tanβ are in AP
tanα+tanβ=2a ...(2)
Also, given γ=α+β
tanγ=tan(α+β)
tanγ=tanα+tanβ1tanαtanβ
22=2a1b2 (by (1) and (2))
a3(1b2)3=113=1331.

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