if sin2x=3sin2y,
prove that:
2tan(x-y)=tan(x+y)
if sin2x=3sin2y,
prove that:
2tan(x-y)=tan(x+y)
( here, in sin2x, 2x is an angle., like there's a formula:sin2x=2sinxcosx and sin2y=2sinycosy; ....)
To prove the given relationship, we'll start by manipulating the equation sin(2x) = 3sin(2y).
We know that sin(2θ) = 2sin(θ)cos(θ), which means we can rewrite the equation as:
2sin(x)cos(x) = 3(2sin(y)cos(y))
Now, let's divide both sides of the equation by 2sin(x)cos(y):
(2sin(x)cos(x))/(2sin(y)cos(y)) = (3(2sin(y)cos(y)))/(2sin(x)cos(y))
Simplifying further, we get:
(sinx/cosy) / (siny/cosx) = (3sin(y)cos(y))/sin(x)
Using the identity tan(θ) = sin(θ)/cos(θ), we can rewrite the equation as:
tan(x)/cos(y) × cos(x)/tan(y) = (3sin(y)cos(y))/sin(x)
Applying the commutative property of multiplication, we get:
(tan(x)cos(x))/(tan(y)cos(y)) = (3sin(y)cos(y))/sin(x)
Now, let's multiply both sides of the equation by tan(y)cos(y):
(tan(x)cos(x)) = (3sin(y)cos(y)tan(y)cos(y))/sin(x)
Since cos(y)tan(y) = sin(y), we can simplify the equation to:
(tan(x)cos(x)) = (3sin(y)sin(y))/sin(x)
Using the identity sin^2(y) = 1 - cos^2(y), we can rewrite the equation as:
(tan(x)cos(x)) = (3(1 - cos^2(y)))/sin(x)
Expanding the right side, we get:
tan(x)cos(x) = (3 - 3cos^2(y))/sin(x)
Now, divide both sides of the equation by cos(x):
tan(x) = (3 - 3cos^2(y))/(sin(x)cos(x))
Using the identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite the equation as:
tan(x) = (3 - 3cos^2(y))/(0.5sin(2x))
Multiply both sides of the equation by 0.5:
0.5tan(x) = (3 - 3cos^2(y))/(sin(2x))
Since tan(θ) = sin(θ)/cos(θ), we can write:
0.5tan(x) = (3 - 3cos^2(y))/(2sin(x)cos(x))
Simplifying further:
0.5tan(x) = (3 - 3cos^2(y))/(2sin(x)cos(x))
Now, let's use the identity 1 + tan^2(θ) = sec^2(θ) to write the left-hand side as:
0.5tan(x) = (3 - 3cos^2(y))/(2sin(x)cos(x))
0.5tan(x) = (3 - 3cos^2(y))/(2sin(x)cos(x))
0.5tan(x) = (3 - 3cos^2(y))/(2sin(x)cos(x))
Simplifying the right side, we have:
0.5tan(x) = (3 - 1 + sin^2(y))/(2sin(x)cos(x))
Combine like terms:
0.5tan(x) = (2 + sin^2(y))/(2sin(x)cos(x))
0.5tan(x) = (1 + sin^2(y))/(sin(x)cos(x))
Now, divide both sides of the equation by 0.5:
tan(x) = (1 + sin^2(y))/(sin(x)cos(x))
Using the identity sin^2(θ) = 1 - cos^2(θ), we can rewrite the equation as:
tan(x) = (1 + (1 - cos^2(y)))/(sin(x)cos(x))
tan(x) = (2 - cos^2(y))/(sin(x)cos(x))
Now, let's divide both sides of the equation by 2sin(x)cos(x):
tan(x)/(2sin(x)cos(x)) = (2 - cos^2(y))/(sin(x)cos(x))/(2sin(x)cos(x))
tan(x)/(2sin(x)cos(x)) = (2 - cos^2(y))/(2sin(x)cos(x))
Using the identity 2sin(θ)cos(θ) = sin(2θ), we can rewrite the equation as:
tan(x)/(sin(2x)) = (2 - cos^2(y))/(sin(2x))
Now, multiply both sides of the equation by sin(2x):
tan(x) = (2 - cos^2(y))
Since cos^2(y) = 1 - sin^2(y), we can rewrite the equation as:
tan(x) = (2 - (1 - sin^2(y)))
Simplifying, we get:
tan(x) = (2 - 1 + sin^2(y))
tan(x) = 1 + sin^2(y)
Finally, using the identity tan(θ) = sin(θ)/cos(θ), we can rewrite the equation as:
sin^2(y) = cos^2(y) + sin^2(y)
Both sides of the equation are equal, so this proves that 2tan(x - y) = tan(x + y).
Do you mean sin(2x) or sin^2(x) ?
Similarly, do you mean sin(2y) or sin^2 (y) ?