Question

If `0 < β < α < π/4, cos (α + β) = 3/5` and cos (α – β) = `4/5`, then sin 2α is equal to ______.

पर्याय

• 0

• 1

• 2

• None of these

Answer 1

If `0 < β < α < π/4, cos (α + β) = 3/5` and cos (α – β) = `4/5`, then sin 2α is equal to 1.

Explanation:

sin 2α = sin {(α + β) + (α – β)}

= sin (α + β) cos (α – β) + cos (α + β) sin (α – β)  ...(i)

Now, sin (α + β) = `sqrt(1 - cos^2 (α + β))`

= `sqrt(1 - 9/25)`   ...`[∵ 0 < α + β < π/2]`

= `4/5` 

and sin (α – β) = `sqrt(1 - cos^2 (α - β))`

= `sqrt(1 - 16/25)`  ...`[∵ 0 < α - β < π/4]`

= `3/5` 

From equation (i),

sin 2α = `4/5 . 4/5 + 3/5 . 3/5`

= `16/25 + 9/25`

= `25/25`

= 1