To simplify the given expression, we can manipulate the trigonometric identities and equations.
Given: sin^2x - 3cos^2x + 2sinx + cosx = 0
We know that sin^2x + cos^2x = 1
Rewriting the given expression by substituting cos^2x = 1 - sin^2x, we get:
sin^2x - 3(1 - sin^2x) + 2sinx + cosx = sin^2x - 3 + 3sin^2x + 2sinx + cosx = 4sin^2x + 2sinx + cosx - 3 = 0
Now, we can substitute sinx = t to simplify the expression:
4t^2 + 2t + √(1 - t^2) - 3 = 0
This is the simplified form of the given trigonometric expression.
To simplify the given expression, we can manipulate the trigonometric identities and equations.
Given: sin^2x - 3cos^2x + 2sinx + cosx = 0
We know that sin^2x + cos^2x = 1
Rewriting the given expression by substituting cos^2x = 1 - sin^2x, we get:
sin^2x - 3(1 - sin^2x) + 2sinx + cosx =
sin^2x - 3 + 3sin^2x + 2sinx + cosx =
4sin^2x + 2sinx + cosx - 3 = 0
Now, we can substitute sinx = t to simplify the expression:
4t^2 + 2t + √(1 - t^2) - 3 = 0
This is the simplified form of the given trigonometric expression.