Proving The Pythagorean Trig Identity With Cot: Derivation And Applications

Pythagorean Trig Identity with cot

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

The Pythagorean Trig Identity is a fundamental formula in trigonometry that relates the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) to each other. The identity is named after Pythagoras, the famous Greek mathematician who studied the relationship between the sides of a right triangle.

The Pythagorean Trig Identity involving cotangent is:

cot²θ + 1 = csc²θ

where cotθ is the ratio of the adjacent side to the opposite side of a right triangle, and cscθ is the reciprocal of the sine function. The identity can be derived using the Pythagorean Theorem and the definitions of the trigonometric functions.

To prove the identity, start with the Pythagorean Theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse:

a² + b² = c²

where a and b are the lengths of the legs, and c is the length of the hypotenuse. Dividing both sides by c² gives:

a²/c² + b²/c² = 1

Recall that:

cotθ = a/b

cscθ = 1/sinθ = c/h

where h is the length of the hypotenuse. Therefore:

(a/b)² + 1 = (c/h)²

Simplifying:

a²/b² + 1 = c²/h²

Substituting a² + b² = c²:

a²/b² + 1 = (a² + b²)/h²

Multiplying both sides by b²:

a² + b² = b²(a²/b² + 1) = b²(c²/h²)

Substituting c/h = 1/sinθ:

a² + b² = b²(c²sin²θ/h²sin²θ) = b²/cos²θ

Dividing by b²cos²θ:

(a/b)² + 1/cos²θ = 1/sin²θ

Recalling that cot²θ = (a/b)² and csc²θ = 1/sin²θ:

cot²θ + 1 = csc²θ

Therefore, the Pythagorean Trig Identity with cot is proved.

This identity is useful in trigonometry to solve problems involving right triangles and trigonometric functions. For example, given the length of the adjacent side and the angle of elevation, we can use the identity to find the length of the hypotenuse or the opposite side. It is also used in calculus and physics to evaluate integrals and solve differential equations involving trigonometric functions.

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