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Question

If tanβ=2sinαsinγsin(α+γ) show that tanα,tanβ,tanγ are in H.P. where α0andγ0andβπ2

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Solution

We have,
tanβ=2sinαsinγsin(α+γ)

tanβ=2sinαsinγsinαcosγ+cosαsinγ

tanβ=2sinαsinγcosαcosγsinαcosγ+cosαsinγcosαcosγ

tanβ=2tanαtanγsinαcosγcosαcosγ+cosαsinγcosαcosγ

tanβ=2tanαtanγsinαcosα+sinγcosγ

tanβ=2tanαtanγtanα+tanγ

So,
tanα,tanβ,tanγ are in H.P

Hence, proved.

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