if sin2x=3sin2y, find 2tan(x-y)
To find 2tan(x-y), we first need to find the values of x and y.
Given that sin2x = 3sin2y, we can use the trigonometric identity:
sin2θ = 2sinθcosθ
Applying this identity, we have:
2sinxcox = 3 * 2sinycoy
sinxcox = 3sinycosy
We know that tanθ = sinθ / cosθ. So we can express sinx / cosx and siny / cosy in terms of tanx and tany:
For sinxcox = 3sinycosy:
sinxcox = 3 * siny * cosy
sinx = 3tanycosy
Dividing by cosx, we get:
tanx = 3tany
Since tan2θ = 2tanθ / (1 - tan²θ), we can use this identity to find tan(x-y):
tan(x-y) = (tanx - tany) / (1 + tanxtany)
= (3tany - tany) / (1 + 3tanytany)
= 2tany / (1 + 3tany²)
Finally, to find 2tan(x-y), we multiply tan(x-y) by 2:
2tan(x-y) = 2 * [2tany / (1 + 3tany²)]
= 4tany / (1 + 3tany²)
Therefore, 2tan(x-y) = 4tany / (1 + 3tany²).