Simplifying the Expression 1 + cot^2(x) | Working with Tangents, Cotangents, and Secants

1+cot^2 x=

To find the value of the expression 1 + cot^2(x), we can start by recognizing the relationship between cotangent (cot) and tangent (tan)

To find the value of the expression 1 + cot^2(x), we can start by recognizing the relationship between cotangent (cot) and tangent (tan). Since cot(x) is the reciprocal of tan(x), we can rewrite the expression using the Pythagorean identity for tangent:

tan^2(x) + 1 = sec^2(x),

where sec(x) represents the secant function.

Now, substituting cot(x) for 1/tan(x), we get:

(1/tan(x))^2 + 1 = sec^2(x).

Next, by simplifying the left-hand side of the equation, we have:

(1/tan(x))^2 + 1 = (1/tan^2(x)) + 1 = (1 + tan^2(x))/tan^2(x).

Now, we can substitute tan^2(x) for sec^2(x) – 1 in the equation:

(1 + tan^2(x))/tan^2(x) = (sec^2(x) – 1 + tan^2(x))/tan^2(x).

Further simplifying the numerator, we have:

(sec^2(x) + tan^2(x) – 1)/tan^2(x).

Finally, using the Pythagorean identity sec^2(x) = 1 + tan^2(x), we get:

(sec^2(x) + tan^2(x) – 1)/tan^2(x) = (1 + tan^2(x) + tan^2(x) – 1)/tan^2(x) = 2(tan^2(x))/tan^2(x) = 2.

Therefore, 1 + cot^2(x) = 2.

In summary, the expression 1 + cot^2(x) simplifies to 2.

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