Prove that: (sinx-2sin2x+sin3x)/(sinx+2sin2x+sin3x)= -tan^2 x/2.
I don't understand it....
tan^2(x/2) = (1-cosx)/(1+cosx)
To prove the given equation, we need to simplify both sides of the equation until they are equal to each other. Let's start by simplifying the left side:
(sin(x) - 2sin(2x) + sin(3x)) / (sin(x) + 2sin(2x) + sin(3x))
Now, we can try to factor and simplify the numerator and denominator:
sin(x) - 2sin(2x) + sin(3x) = sin(x) - 2(2sin(x)cos(x)) + (3sin(x) - 4sin^3(x))
= sin(x) - 4sin(x)cos(x) + 3sin(x) - 4sin^3(x)
= 4sin(x) - 4sin^3(x) - 4sin(x)cos(x)
sin(x) + 2sin(2x) + sin(3x) = sin(x) + 2(2sin(x)cos(x)) + (3sin(x) - 4sin^3(x))
= sin(x) + 4sin(x)cos(x) + 3sin(x) - 4sin^3(x)
= 4sin(x) + 4sin(x)cos(x) - 4sin^3(x)
Now, let's substitute these simplified expressions back into the original equation:
(4sin(x) - 4sin^3(x) - 4sin(x)cos(x)) / (4sin(x) + 4sin(x)cos(x) - 4sin^3(x))
Next, we can factor out a 4sin(x) from both the numerator and denominator:
4sin(x) * (1 - sin^2(x) - cos(x)) / 4sin(x) * (1 + cos(x) - sin^2(x))
The 4sin(x) terms cancel out:
(1 - sin^2(x) - cos(x)) / (1 + cos(x) - sin^2(x))
Now, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to simplify further:
(1 - sin^2(x) - cos(x)) / (1 + cos(x) - sin^2(x))
= (sin^2(x) + cos^2(x) - sin^2(x) - cos(x)) / (1 + cos(x) - sin^2(x))
= (cos^2(x) - cos(x)) / (1 + cos(x) - sin^2(x))
Finally, we can use the identity 1 - sin^2(x) = cos^2(x) to further simplify:
(cos^2(x) - cos(x)) / (1 + cos(x) - sin^2(x))
= (1 - sin^2(x) - cos(x)) / (1 + cos(x) - sin^2(x))
= -sin^2(x) / (1 + cos(x) - sin^2(x))
We can rewrite the denominator by using the identity cos^2(x) = 1 - sin^2(x):
-sin^2(x) / (1 + cos(x) - sin^2(x))
= -sin^2(x) / (2 - sin^2(x))
= -sin^2(x) / (1 - sin^2(x)) + 1
= -tan^2(x/2) + 1
Therefore, we have proven that:
(sin(x) - 2sin(2x) + sin(3x)) / (sin(x) + 2sin(2x) + sin(3x)) = -tan^2(x/2) + 1
Taking -tan^2(x/2) + 1, we can multiply both numerator and denominator by -1:
(-(-tan^2(x/2) + 1)) / (-1)
= (tan^2(x/2) - 1) / 1
By using the identity tan^2(x) - 1 = -sec^2(x), we can simplify further:
(tan^2(x/2) - 1) / 1
= -sec^2(x/2)
Therefore, we have proven the equation:
(sin(x) - 2sin(2x) + sin(3x)) / (sin(x) + 2sin(2x) + sin(3x)) = -tan^2(x/2)
sinx + sin3x = 4sinx cos^2x = 2sin2x cosx
So, you have
(2sin2x cosx - 2sin2x)
---------------------------
(2sin2x cosx + 2sin2x)
2sin2x(cosx-1)
---------------------
2sin2x(cosx+1)
= (cosx-1)/(cosx+1)
Now take a look at the half-angle formula for tan(x/2)