Final answer:
The trigonometric identity is verified using basic identities; the terms simplify to cos²(x) and sin²(x), which according to the Pythagorean identity sum up to 1.
Step-by-step explanation:
To verify the trigonometric identity sin² x · cot² x + cos² x · tan² x=1, we can use fundamental trigonometric identities and properties.
First, we know that cot(x) = 1/tan(x) and tan(x) = sin(x)/cos(x). Using these, we can rewrite the first part of the equation as sin²(x) · (1/tan(x))² which simplifies to sin²(x) · cos²(x)/sin²(x) because cot(x) is the reciprocal of tan(x). This simplifies further to cos²(x).
Similarly, we can rewrite the second part of the equation as cos²(x) · (sin(x)/cos(x))² which simplifies to cos²(x) · sin²(x)/cos²(x). This simplifies to sin²(x). Therefore, the equation becomes cos²(x) + sin²(x). According to the Pythagorean identity, cos²(x) + sin²(x) = 1. Hence, the identity is verified.