Question: Verify the identity.
sin² x cot² x + sin² x = 1
To verify that an equation is an identity, start with the more complicated side and transform the side of the equation into the other side by a sequence of steps, each of which produces an identity.
sin² x cot² x + sin² x
1. **State the problem:** Verify the identity $$\sin^2 x \cot^2 x + \sin^2 x = 1$$.
2. **Start with the left side (LHS):** $$\sin^2 x \cot^2 x + \sin^2 x$$.
3. **Rewrite cotangent in terms of sine and cosine:** $$\cot x = \frac{\cos x}{\sin x}$$, so
$$\sin^2 x \cot^2 x + \sin^2 x = \sin^2 x \left(\frac{\cos^2 x}{\sin^2 x}\right) + \sin^2 x$$.
4. **Simplify by canceling common factors:**
$$= \cancel{\sin^2 x} \frac{\cos^2 x}{\cancel{\sin^2 x}} + \sin^2 x = \cos^2 x + \sin^2 x$$.
5. **Use the Pythagorean identity:**
$$\cos^2 x + \sin^2 x = 1$$.
6. **Conclusion:** The left side simplifies to the right side, so the identity is verified.
**Final answer:** $$\sin^2 x \cot^2 x + \sin^2 x = 1$$ is true.