What is equal to 1 plus cot squared?

To understand why the correct answer is related to (1 + \cot^2), it's important to remember key trigonometric identities. The identity that relates cotangent and cosecant is:

[

1 + \cot^2(\theta) = \csc^2(\theta)

]

This identity can be derived from the Pythagorean identity. Here’s a breakdown:

1. Cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle, or (\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}).

2. Squaring cotangent gives (\cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)}).

3. Now, if you add 1 (which can be thought of as (\frac{\sin^2(\theta)}{\sin^2(\theta)})) to (\cot^2(\theta)):

[

1 + \cot^2(\theta) = \frac{\sin^2(\theta) + \cos^2(\theta)}{\sin^2(\theta)}

]

4. Using the fundamental Pythagorean identity