We are asked to simplify the expression $sin7x + sinx - 2sin2xcos3x$. We need to choose the correct simplified expression from the options given.

1. Problem Description

We are asked to simplify the expression $sin7x + sinx - 2sin2xcos3x$. We need to choose the correct simplified expression from the options given.

2. Solution Steps

We can use the sum-to-product formula for $sin7x + sinx$:

$sinA + sinB = 2sin(\frac{A+B}{2})cos(\frac{A-B}{2})$

In our case, $A = 7x$ and $B = x$. Therefore,

$sin7x + sinx = 2sin(\frac{7x+x}{2})cos(\frac{7x-x}{2}) = 2sin(\frac{8x}{2})cos(\frac{6x}{2}) = 2sin4xcos3x$.

Now, we substitute this back into the original expression:

$sin7x + sinx - 2sin2xcos3x = 2sin4xcos3x - 2sin2xcos3x$

We can factor out $2cos3x$:

$2cos3x(sin4x - sin2x)$.

We can use the sum-to-product formula for $sin4x - sin2x$:

$sinA - sinB = 2cos(\frac{A+B}{2})sin(\frac{A-B}{2})$

In our case, $A = 4x$ and $B = 2x$. Therefore,

$sin4x - sin2x = 2cos(\frac{4x+2x}{2})sin(\frac{4x-2x}{2}) = 2cos(\frac{6x}{2})sin(\frac{2x}{2}) = 2cos3xsinx$.

Now, substitute this back into the expression:

$2cos3x(sin4x - sin2x) = 2cos3x(2cos3xsinx) = 4cos^2(3x)sinx$.

3. Final Answer

The simplified expression is $4cos^2(3x)sinx$. Therefore, the correct answer is option c.