Question

If cos (α + β) = `4/5` and sin (α - β) = `5/13` where (α + β) and (α - β) are acute, then find tan 2α.

Answer 1

cos (α + β) = `4/5`

sin (α + β) = `3/5`

tan (α + β) = `3/4`

sin (α - β) = `5/13`

cos (α - β) = `12/13`

tan (α - β) = `5/12`

Now tan 2α = tan [(α + β) + (α - β)]

`= (tan (α + β) + (α - β))/(1 - tan (α + β) tan(α - β))`

`[tan (x + y) = (tan x + tan y)/(1 - tan x tan y)]`

`= (3/4 + 5/12)/(1 - (3/4)(5/12))`

`= ((9 + 5)/12)/((48 - 15)/48)`

`= (14/12)/(33/48)`

`= 14/12 xx 48/33`

`= (14 xx 4)/33`

`therefore tan (2alpha) = 56/33`