Final answer:
The expression (csc²(x) - cot²(x)) / csc²(x) simplifies to 1 using trigonometric identities and the Pythagorean identity.
Step-by-step explanation:
The expression (csc²(x) - cot²(x)) / csc²(x) can be simplified using trigonometric identities. Recall that the cosecant function is the reciprocal of the sine function (csc(x) = 1/sin(x)), and the cotangent function is the reciprocal of the tangent function (cot(x) = 1/tan(x)) or can also be expressed as cot(x) = cos(x)/sin(x). To simplify the original expression, we can express cot²(x) in terms of sine and cosine as follows:
cot²(x) = cos²(x) / sin²(x).
Putting this into our original expression gives us:
(1/sin²(x) - cos²(x) / sin²(x)) / (1/sin²(x)). This simplifies to 1 - cos²(x) / sin²(x) after we combine the terms in the numerator. Using the Pythagorean identity sin²(x) + cos²(x) = 1 , we can replace 1 - cos²(x) with sin²(x), leaving us with:
sin²(x) / sin²(x), which simplifies further to just 1. Hence, the simplified form of the given expression is 1.