Please prove:
2sinx+sin2x = 2sin^3(x)/1-cosx
To prove the equality, we need to simplify each side of the equation and show that they are equal.
Starting with the left side, we have:
2sinx + sin2x
Using the double-angle formula for sine (sin2x = 2sinx*cosx), we can rewrite the expression as:
2sinx + 2sinx*cosx
Factoring out sinx, we get:
sinx(2 + 2cosx)
Now, moving to the right side of the equation, we have:
2sin^3(x) / (1 - cosx)
Using the identity sin^2(x) = 1 - cos^2(x), we can rewrite the expression as:
2sin^3(x) / sin^2(x)
Since sin^2(x) is the denominator, we can rewrite the expression as:
2sin(x) * sin(x) / sin^2(x)
Now, canceling out sin(x) in the numerator and denominator, we get:
2sin(x) / 1
Simplifying further, we have:
2sin(x)
Comparing this expression to the simplified left side, we can see that they are equal:
2sin(x) + sin(x)(2 + 2cos(x)) = 2sin(x)
Hence, the left side equals the right side, and the equation is proven.
To prove the given equation:
2sin(x) + sin(2x) = 2sin^3(x) / (1 - cos(x))
we can start by simplifying each side of the equation separately.
Let's start with the left side of the equation:
2sin(x) + sin(2x)
Using the double angle formula for sin(2x), which states that sin(2x) = 2sin(x)cos(x), we can substitute and simplify:
2sin(x) + 2sin(x)cos(x)
Next, let's focus on the right side of the equation:
2sin^3(x) / (1 - cos(x))
To simplify this, we need to work on the numerator and the denominator separately.
Numerator:
To simplify the numerator, we can use the identity sin^2(x) = 1 - cos^2(x):
sin^2(x) = 1 - cos^2(x)
sin^2(x) = (1 - cos(x))(1 + cos(x))
sin^2(x) = 1 - cos(x)
If we substitute this simplification into the numerator, we get:
2(1 - cos(x))
Denominator:
No simplification is needed for the denominator.
Now, if we substitute the simplified numerator and unchanged denominator into the original equation, we get:
2sin(x) + 2sin(x)cos(x) = 2(1 - cos(x)) / (1 - cos(x))
Next, let's simplify the equation even further.
Starting with the left side of the equation, we can factor out a common term of 2sin(x):
2sin(x) + 2sin(x)cos(x) = 2sin(x)(1 + cos(x))
Now, we can see that (1 + cos(x)) is a common factor in both the numerator and the denominator of the right side of the equation.
So, we can cancel out this common factor:
2sin(x)(1 + cos(x)) = 2(1 - cos(x)) / (1 - cos(x))
2sin(x) = 2 / (1 - cos(x))
Finally, we can simplify further by canceling out the 2's on both sides of the equation:
sin(x) = 1 / (1 - cos(x))
And there you have it! We have proved that:
2sin(x) + sin(2x) = 2sin^3(x) / (1 - cos(x))
can be simplified to:
sin(x) = 1 / (1 - cos(x))
through a series of algebraic steps.