Simplify the expression.sin2x + sinx + cos 2x - 1

To simplify the expression sin2x + sinx + cos2x - 1, we can use trigonometric identities to simplify each term individually.

1. Let's start with sin2x. Using the double angle identity for sine, we have sin2x = 2sinxcosx.

2. Next, we have sinx, which cannot be simplified further.

3. Moving on to cos2x, we can use the double angle identity for cosine to simplify it as well. The identity is cos2x = cos^2x - sin^2x. Since sin^2x + cos^2x = 1 (from the Pythagorean identity), we can rewrite cos2x as cos^2x - (1 - cos^2x) = 2cos^2x - 1.

4. Lastly, we have -1, which is already simplified.

Now, let's substitute the simplified forms back into the expression:

2sinxcosx + sinx + 2cos^2x - 1.

We can combine like terms:

sinx(2cosx + 1) + 2cos^2x - 1.

And that is the simplified form of the expression sin2x + sinx + cos2x - 1.