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.