1 + cot^2x =
csc^2x
There are a few ways to approach this problem, but one common method is to use the identity: 1 + cot^2 x = csc^2 x.
To prove this identity, we can start with the Pythagorean identity:
sin^2 x + cos^2 x = 1
Dividing both sides by sin^2 x, we get:
1 + cos^2 x / sin^2 x = 1 / sin^2 x
Using the definition of cotangent (cot x = cos x / sin x), we can substitute cos^2 x / sin^2 x as cot^2 x + 1:
1 + cot^2 x + 1 = (1 + cos^2 x) / sin^2 x
Simplifying the right side using the Pythagorean identity, we get:
2 = csc^2 x
Therefore, 1 + cot^2 x = csc^2 x.
So for any value of x, 1 + cot^2x is equal to csc^2 x.
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