To prove this identity, we can start by multiplying both sides of the equation by cot^2(x) to get rid of the cot^2(x) on the left-hand side:
(1+tan^2(x))cot^2(x) = csc^2(x) * cot^2(x)
1 + tan^2(x) * cot^2(x) = csc^2(x) * cot^2(x)
Next, we can use the identity cot^2(x) = csc^2(x) - 1 to simplify the left-hand side:
1 + (tan^2(x) * cot^2(x)) = csc^2(x) * cot^2(x)
1 + (tan^2(x) * (csc^2(x) - 1)) = csc^2(x) * (csc^2(x) - 1)
We can simplify the right-hand side by multiplying the two factors in the parentheses:
1 + (tan^2(x) * (csc^2(x) - 1)) = (csc^2(x))^2 - csc^2(x)
Now we can use the identity tan^2(x) = 1 / csc^2(x) to simplify the left-hand side:
1 + (1/csc^2(x) * (csc^2(x) - 1)) = (csc^2(x))^2 - csc^2(x)
1 + (1 - 1/csc^2(x)) = (csc^2(x))^2 - csc^2(x)
Next, we can combine like terms on both sides of the equation to get:
1 = (csc^2(x))^2 - csc^2(x)
1 = csc^4(x) - csc^2(x)
Finally, we can add csc^2(x) to both sides to get:
1 + csc^2(x) = csc^4(x)
csc^2(x) = csc^4(x) - 1
This is the same as the original equation, so we have proven that the identity is true.