Simplifying the Expression 1 + cot^2(x) Using Trigonometric Identities

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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