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Question

If tanα=asinβ1acosβandtanβ=bsinα1bcosα then show that sinαsinβ=ab.

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Solution

We have,

tanα=asinβ1acosβ

sinαcosα=asinβ1acosβ

sinαasinαcosβ=acosαsinβ

sinα=a(sinαcosβ+cosαsinβ)

sinα=asin(α+β) ……… (1)

Similarly,

tanβ=bsinα1bcosα

sinβcosβ=bsinα1bcosα

sinβbcosαsinβ=bsinαcosβ

sinβ=b(sinαcosβ+cosαsinβ)

sinβ=bsin(α+β) …….. (2)

From equation (1) and (2), we get

sinαsinβ=ab

Hence, proved.


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