1 + cot^2(x)
To simplify the expression 1 + cot^2(x), we need to use trigonometric identities
To simplify the expression 1 + cot^2(x), we need to use trigonometric identities.
Recall that the cotangent function is the reciprocal of the tangent function. It can be written as cot(x) = 1/tan(x).
Using this identity, we can rewrite cot^2(x) as (1/tan(x))^2.
To simplify further, we need to use the Pythagorean identity: tan^2(x) + 1 = sec^2(x).
Rearranging this identity, we have sec^2(x) – tan^2(x) = 1.
Now, let’s substitute sec^2(x) – tan^2(x) for 1:
1 + cot^2(x) = 1 + (1/tan(x))^2
Since cot(x) = 1/tan(x), we can substitute cot(x) into the expression:
1 + cot^2(x) = 1 + (1/cot(x))^2
Now, let’s substitute the reciprocal of cot(x) to get:
1 + cot^2(x) = 1 + (cot(x))^(-2)
Using the power of a reciprocal property, we can rewrite (cot(x))^(-2) as 1/(cot(x))^2:
1 + cot^2(x) = 1 + 1/(cot(x))^2
Now, let’s substitute the reciprocal of cot(x) again:
1 + cot^2(x) = 1 + 1/(1/tan(x))^2
Simplifying the expression (1/tan(x))^2, we have:
1 + cot^2(x) = 1 + 1/(1/tan(x))^2
= 1 + 1/(1^2/tan^2(x))
= 1 + tan^2(x)
Therefore, the simplified expression for 1 + cot^2(x) is tan^2(x) + 1.
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