1. **State the problem:** Solve the trigonometric equation $$\sin^2 x - \cos x - 1 = 0$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x = 1 - \cos^2 x$$. 3. **Substitute** this into the equation: $$1 - \cos^2 x - \cos x - 1 = 0$$ which simplifies to $$-\cos^2 x - \cos x = 0$$. 4. **Multiply both sides by -1** to simplify: $$\cos^2 x + \cos x = 0$$. 5. **Factor the expression:** $$\cos x (\cos x + 1) = 0$$. 6. **Set each factor equal to zero:** - $$\cos x = 0$$ - $$\cos x + 1 = 0 \Rightarrow \cos x = -1$$. 7. **Solve for x:** - For $$\cos x = 0$$, solutions are $$x = \frac{\pi}{2} + k\pi$$, where $$k$$ is any integer. - For $$\cos x = -1$$, solution is $$x = \pi + 2k\pi$$, where $$k$$ is any integer. **Final answer:** $$x = \frac{\pi}{2} + k\pi \quad \text{or} \quad x = \pi + 2k\pi, \quad k \in \mathbb{Z}$$.