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Question

If tanα=asinβ1acosβ and tanβ=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=bsin(α+β)........(2)

On dividing equation (1) from (2), we get

sinαsinβ=asin(α+β)bsin(α+β)

sinαsinβ=ab

Hence, proved.


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