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If m sin θ = n sin (θ + 2α), then prove that tan (θ + α) cot α = (m + n)/(m – n).

[Hints: Express sin(θ + 2α) / sinθ = m/n and apply componendo and dividend]

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Best answer

According to the question,

m sin θ = n sin (θ + 2α)

To prove:

tan (θ + α)cot α =(m + n)/(m – n)

Proof:

m sin θ = n sin (θ + 2α)

⇒ sin(θ + 2α) / sinθ = m/n

Applying componendo-dividendo rule, we have,

By transformation formula of T-ratios,

We know that,

sin A + sin B = 2 sin ((A+B)/2) cos ((A – B)/2)

And,

sin A – sin B = 2 cos ((A+B)/2) sin ((A – B)/2)

On applying the formula, we get,

Therefore, tan (θ + α) cot α = (m + n)/(m – n)

Hence Proved

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