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Question
If cos(α + β) = `4/5` and sin(α – β) = `5/13`, where α lie between 0 and `pi/4`, find the value of tan2α.
[Hint: Express tan2α as tan(α + β + α – β)]
Solution
Given that: cos(α + β) = `4/5`
∴ tan(α + β) = `3/4`
And sin(α – β) = `5/13`
∴ tan(α – β) = `5/12`
Now tan 2α = tan[α + β + α – β]
= tan[(α + β) + (α – β)]
= `(tan(alpha + beta) + tan(alpha - beta))/(1 - tan(alpha + beta).tan(alpha - beta))`
= `(3/4 + 5/12)/(1 - 3/4 xx 5/12)`
= `((9 + 5)/12)/((48 - 15)/48)`
= `14/12 xx 48/33`
= `56/33`
Hence, tan 2α = `56/33`.
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