Understanding the Relationship between Cotangent and the Pythagorean Identity: Expanding on the Pythagorean Identity to Find the Relationship between Cotangent and Cosecant

Pythagorean Identity: cot^2(x) + 1 =

The Pythagorean Identity is a fundamental trigonometric identity that relates the values of the three primary trigonometric functions – sine, cosine, and tangent – in terms of each other

The Pythagorean Identity is a fundamental trigonometric identity that relates the values of the three primary trigonometric functions – sine, cosine, and tangent – in terms of each other.

The Pythagorean Identity is given by:

sin^2(x) + cos^2(x) = 1

This identity represents the relationship between the lengths of the sides of a right triangle and is based on the Pythagorean theorem. It shows that the square of the sine of an angle plus the square of the cosine of the same angle will always equal 1.

Now, let’s consider the reciprocal of the tangent function, which is the cotangent function (cot(x)). The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right triangle. The cotangent function is the multiplicative inverse or reciprocal of the tangent function:

cot(x) = 1 / tan(x)

To find the relationship between cotangent and the Pythagorean Identity, we can substitute the definition of tangent into the equation:

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

Using the definition of tangent and simplifying further:

cot^2(x) + 1 = (1 / (sin(x) / cos(x)))^2 + 1
= (cos^2(x) / sin^2(x)) + 1

Now, the numerator of the fraction can be written using the Pythagorean Identity as:

cos^2(x) = 1 – sin^2(x)

Substituting this value back into the equation, we have:

cot^2(x) + 1 = ((1 – sin^2(x)) / sin^2(x)) + 1
= 1/sin^2(x) – sin^2(x)/sin^2(x) + 1
= (1 – sin^2(x) + sin^2(x))/sin^2(x) + 1
= 1/sin^2(x) + 1

This simplifies to:

cot^2(x) + 1 = csc^2(x)

Therefore, the Pythagorean Identity for the cotangent function is:

cot^2(x) + 1 = csc^2(x)

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