The Pythagorean Identity For Csc And Cot Functions In Trigonometry

Pythagorean Identity (csc and cot)

cot²θ+1 = csc²θ

The Pythagorean identity for csc and cot functions relates the values of these trigonometric functions to the values of sine and cosine functions. It is given as:

csc²θ = 1 + cot²θ

where θ is the angle in radians.

This identity can also be written in terms of sine and cosine functions as:

1 + tan²θ = sec²θ

Both of these forms of the pythagorean identity relate the values of three of the six trigonometric functions to the values of the remaining functions.

To understand the Pythagorean identity for csc and cot, we need to first understand what these functions represent.

Cosecant function (csc) is the reciprocal of the sine function. It represents the ratio of the length of the hypotenuse to the length of the opposite side of a right triangle.

Cotangent function (cot) is the reciprocal of the tangent function. It represents the ratio of the adjacent side of a right triangle to the opposite side of the triangle.

The Pythagorean identity for csc and cot derives from the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. In other words,

a² + b² = c²

where a and b are the lengths of the legs, and c is the length of the hypotenuse.

If we divide both sides of this equation by c², we get:

a²/c² + b²/c² = 1

Now, we can substitute the values of sine and cosine in terms of the legs and hypotenuse of the triangle, using the definitions of these trigonometric functions:

sinθ = b/c

cosθ = a/c

Substituting these values, we get:

sin²θ + cos²θ = 1

This is the Pythagorean identity for sine and cosine functions. Now, if we take the reciprocal of both sides of this equation, we get:

csc²θ + cot²θ = 1

which is the Pythagorean identity for csc and cot functions.

In practical applications, the Pythagorean identity for csc and cot is used to simplify complex trigonometric expressions involving these functions. For example, if we have an expression like:

csc²θ – 1 = cot²θ

we can use the Pythagorean identity to simplify it as:

1 + cot²θ – 1 = cot²θ

cot²θ = cot²θ

which is a true statement. By using this identity, we can avoid tedious and error-prone algebraic manipulations, and arrive at simple and elegant solutions.

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