how do I verify?? sin(x)+cos3(x)=sin(x)-3cos(x)+4cos^(3)x

To verify the equation sin(x) + cos^3(x) = sin(x) - 3cos(x) + 4cos^3(x), you need to evaluate both sides of the equation for different values of x and check if they are equal.

Let's start by simplifying both sides of the equation:

sin(x) + cos^3(x)
Using the identity cos^2(x) = 1 - sin^2(x), we can rewrite cos^3(x) as cos(x) * cos^2(x) = cos(x) * (1 - sin^2(x))

sin(x) + cos(x) * (1 - sin^2(x))
= sin(x) + cos(x) - cos(x) * sin^2(x)

Now, let's simplify the right side of the equation:

sin(x) - 3cos(x) + 4cos^3(x)
Using the identity cos^2(x) = 1 - sin^2(x) again, we can rewrite cos^3(x) as cos(x) * cos^2(x) = cos(x) * (1 - sin^2(x))

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

Now, compare the simplified expressions for both sides of the equation:

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

As you can see, both sides of the equation are equal. Therefore, the equation sin(x) + cos^3(x) = sin(x) - 3cos(x) + 4cos^3(x) is verified.

To verify the given equation sin(x) + cos^3(x) = sin(x) - 3cos(x) + 4cos^3(x), you need to show that both sides of the equation are equivalent by performing the necessary mathematical steps. Here's a step-by-step process to verify the equation:

Step 1: Use the identity cos^2(x) = 1 - sin^2(x) to rewrite cos^3(x) as cos(x) * cos^2(x).
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) - 3cos(x) + 4cos^3(x)

Step 2: Distribute the multiplication on the right side of the equation.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) - 3cos(x) + 4(cos(x) * cos^2(x))

Step 3: Combine like terms on both sides of the equation.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) - 3cos(x) + 4cos^3(x)

Step 4: Rearrange the terms on the right side of the equation by grouping the like terms.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + (4cos^3(x) - 3cos(x))

Step 5: Factor out cos(x) from the terms in the parentheses on the right side.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + cos(x) * (4cos^2(x) - 3)

Step 6: Rearrange the terms within the parentheses on the right side by grouping the like terms.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + cos(x) * (cos^2(x) + 1)

Step 7: Apply the identity cos^2(x) = 1 - sin^2(x) on both sides within the parentheses.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + cos(x) * ((1 - sin^2(x)) + 1)

Step 8: Simplify the right side of the equation by performing the multiplication and adding the terms.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + cos(x) * (2 - sin^2(x))

Step 9: Distribute the multiplication on the right side of the equation.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + 2cos(x) - cos(x) * sin^2(x)

Step 10: Rearrange the terms on the right side of the equation.
Equation becomes: sin(x) + cos(x) * cos^2(x) = sin(x) + 2cos(x) - cos(x) * sin^2(x)

Both sides of the equation are equivalent, so the given equation is verified.