Understanding the Reciprocal Identity for Cotangent: Explained with Examples and Simplification

Reciprocal Identity: cot(x) =

To understand the reciprocal identity for cotangent, we first need to understand the basic definition of cotangent (cot)

To understand the reciprocal identity for cotangent, we first need to understand the basic definition of cotangent (cot).

The cotangent of an angle x is defined as the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, it can be expressed as:

cot(x) = adjacent side / opposite side

Now let’s consider the reciprocal of cotangent. The reciprocal of a number is obtained by taking its multiplicative inverse. Therefore, the reciprocal of cot(x) can be written as:

1 / cot(x)

To simplify the expression further, we need to convert cot(x) into an equivalent fraction with a common denominator. We can express cot(x) as cos(x) / sin(x), where cos(x) represents the cosine of x and sin(x) represents the sine of x.

So, the reciprocal identity for cot(x) becomes:

1 / cot(x) = 1 / (cos(x) / sin(x))

To divide by a fraction, we multiply by its reciprocal. Therefore, we can multiply by the reciprocal of the fraction cos(x) / sin(x), which is sin(x) / cos(x).

1 / cot(x) = 1 / (cos(x) / sin(x)) x (sin(x) / cos(x))

By simplifying the expression further, we can cancel out the common factors:

1 / cot(x) = sin(x) / cos(x)

Finally, we can rewrite sin(x) / cos(x) using the reciprocal identity for tangent (tan):

1 / cot(x) = tan(x)

So, the reciprocal identity for cot(x) is:

cot(x) = 1 / tan(x)

This means that the cotangent of an angle x is equal to the reciprocal of the tangent of the same angle x.

Overall, the reciprocal identity for cot(x) can be represented as cot(x) = 1 / tan(x).

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