Question

If cos (α + β) = 4/5 and sin (α – β) = 5/13 , where α lie between 0 and π/4, find the value of tan2α.

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

Answer 1

According to the question,

cos(α + β) = 4/5 …(i)

We know that,

sin x = √(1 – cos²x)

Therefore,

sin (α + β) = √(1 – cos²(α + β))

⇒ sin (α + β) = √(1 – (4/5)²) = 3/5 …(ii)

Also,

sin(α – β) = 5/13 {given} …(iii)

we know that,

cos x = √(1 – sin²x)

Therefore,

cos (α – β) = √(1 – sin²(α – β))

⇒ cos (α – β) = √(1 – (5/13)²) = 12/13 …(iv)

Therefore,

tan 2α = tan (α + β + α – β)

We know that,

Hence, tan 2α = 56/33