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Question
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
Solution
Given that: `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`
⇒ `(sin(x + y) + sin(x - y))/(sin(x + y) - sin(x - y)) = (a + b + a - b)/(a + b - a + b)` .....(Using componendo and dividendo theorem)
⇒ `(2sin((x + y + x - y)/2) cos ((x + y - x + y)/2))/(2cos((x + y + x - y)/2) sin((x + y - x + y)/2)) = (2a)/(2b)`
⇒ `(sinx . cos y)/(cosx . sin y) = a/b`
⇒ tan x.cot y = `a/b`
⇒ `tanx/tany = a/b`
Hence proved.
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