If where α lie between 0 and π/4, find value of tan 2α

[Hint: Express tan 2α as tan (α + β + α – β]

Given that,

cos(α + β) = 4/5 …(1)

we know that: sin x = √(1 – cos²x)

∴ sin (α + β) = √(1 – cos²(α + β))

⇒ sin (α + β) = …(2)

Also,

sin(α - β) = 5/13 {given} …(3)

we know that: cos x = √(1 – sin²x)

∴ cos (α - β) = √(1 – sin²(α - β))

⇒ cos (α - β) = …(4)

∵ tan 2α = tan (α + β + α – β)

We know that:

∴ tan 2α =

⇒ tan 2α =

Using equation 1,2,3 and 4 we have –

Hence, tan 2α = 56/33