Sin3x=3cos^2xsinx-sin^3x

If you're trying to prove it,

sin(3x) = sin(2x+x)
= sin2x cosx + cos2x sinx
= 2sinx cos^2 x + (cos^2x-sin^2x)sinx
= 2sinx cos^2x + cos^2x sinx - sin^3x
ta da

To solve the equation sin(3x) = 3cos^2(x)sin(x) - sin^3(x), we can use some trigonometric identities to simplify the equation:

Step 1: Apply the identity sin(3x) = 3sin(x) - 4sin^3(x)
The identity sin(3x) = 3sin(x) - 4sin^3(x) can be used to rewrite sin(3x) on the left side of the equation.

So, the equation becomes: 3sin(x) - 4sin^3(x) = 3cos^2(x)sin(x) - sin^3(x)

Step 2: Rearrange the terms
Let's rearrange the equation by bringing all terms to one side and factor out common factors of sin(x):

3sin(x) - 4sin^3(x) - 3cos^2(x)sin(x) + sin^3(x) = 0

Now, we have a polynomial equation in terms of sin(x), so let's continue solving.

Step 3: Factor out sin(x)
Factor out sin(x) from the expression:

sin(x)(3 - 4sin^2(x) - 3cos^2(x) + sin^2(x)) = 0

Step 4: Simplify the expression in the parentheses
Using the identity cos^2(x) + sin^2(x) = 1, we can simplify the expression:

sin(x)(3 - 4sin^2(x) - 3(1 - sin^2(x))) = 0
sin(x)(3 - 4sin^2(x) - 3 + 3sin^2(x)) = 0
sin(x)(-sin^2(x) - 1) = 0

Step 5: Consider the cases separately
Setting each factor to zero gives two possible solutions:

1) sin(x) = 0
2) -sin^2(x) - 1 = 0

Step 6: Solve for sin(x) = 0
The equation sin(x) = 0 has solutions at x = nπ, where n is an integer.

Step 7: Solve for -sin^2(x) - 1 = 0
Rearrange the equation:

-sin^2(x) - 1 = 0
sin^2(x) + 1 = 0

However, there are no real solutions for sin^2(x) + 1 = 0 since sin^2(x) is always non-negative.

So, the solutions to the original equation sin(3x) = 3cos^2(x)sin(x) - sin^3(x) are x = nπ, where n is an integer.

To solve the equation sin(3x) = 3cos^2(x)sin(x) - sin^3(x), we need to simplify both sides of the equation and then find the possible values of x that satisfy the equation. Here's how you can solve it step by step:

Step 1: Use trigonometric identities to simplify the equation:
sin(3x) = 3cos^2(x)sin(x) - sin^3(x)
Using the identity sin(3x) = 3sin(x) - 4sin^3(x), we can rewrite the equation as:
3sin(x) - 4sin^3(x) = 3cos^2(x)sin(x) - sin^3(x)

Step 2: Rearrange the terms:
3sin(x) - 4sin^3(x) - 3cos^2(x)sin(x) + sin^3(x) = 0
Rearranging the terms gives us:
3sin(x) - 3cos^2(x)sin(x) - 4sin^3(x) + sin^3(x) = 0

Step 3: Factor out common terms:
sin(x) (3 - 3cos^2(x) - 4sin^2(x)) + sin^3(x) (1 - 4sin^2(x)) = 0

Step 4: Apply trigonometric identities to further simplify:
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute 1 - cos^2(x) for sin^2(x) in the equation:
sin(x) (3 - 3cos^2(x) - 4(1 - cos^2(x))) + sin^3(x) (1 - 4sin^2(x)) = 0
Simplifying gives us:
sin(x) (3 - 3cos^2(x) - 4 + 4cos^2(x)) + sin^3(x) (1 - 4sin^2(x)) = 0
sin(x) (-1 + cos^2(x)) + sin^3(x) (1 - 4sin^2(x)) = 0

Step 5: Factor out common factors:
- sin(x)(1 - cos^2(x)) + sin^3(x)(1 - 4sin^2(x)) = 0

Step 6: Apply another trigonometric identity:
Using the identity 1 - cos^2(x) = sin^2(x), we can substitute in the equation:
- sin(x)*sin^2(x) + sin^3(x)(1 - 4sin^2(x)) = 0
- sin^3(x) + sin^3(x) - 4sin^5(x) = 0

Step 7: Combine like terms:
- 2sin^3(x) - 4sin^5(x) = 0

Step 8: Factor out a common factor:
- 2sin^3(x)(1 + 2sin^2(x)) = 0

Step 9: Set each factor equal to zero and solve the resulting equations:
sin^3(x) = 0
sin(x) = 0

1 + 2sin^2(x) = 0
2sin^2(x) = -1
sin^2(x) = -1/2

Taking the square root of both sides in sin^2(x) = -1/2 gives us:
sin(x) = +/- sqrt(-1/2)

However, there are no real values for sin(x) where sqrt(-1/2) is defined. So, the only solution is sin(x) = 0.

Step 10: Solve for x:
To find the values of x that satisfy sin(x) = 0, recall that sin(x) = 0 for x = k*pi, where k is an integer.
Therefore, the values of x that satisfy the equation are x = k*pi, where k is an integer.

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