Final answer:
The equation 2cos²x - sinx - 1 = 0 is solved by using the trigonometric identity sin²x + cos²x = 1 to transform it into a quadratic in terms of sinx, and then applying the quadratic formula.
Step-by-step explanation:
To solve the equation 2cos²x - sinx - 1 = 0, we can use a trigonometric identity to transform it into a quadratic equation. The trigonometric identity we can use is sin²x + cos²x = 1, which implies cos²x = 1 - sin²x. Substituting cos²x with 1 - sin²x, we get:
- 2(1 - sin²x) - sinx - 1 = 0
- 2 - 2sin²x - sinx - 1 = 0
- -2sin²x - sinx + 1 = 0
To solve this quadratic equation in terms of sinx, we can use the quadratic formula where a = -2, b = -1, and c = 1. Solving for sinx will give us the possible values for x.