Understanding the Pythagorean Identity: An Exploration of Trigonometric Equations and Applications

Pythagorean identity equal to csc^2(x)

1 + cot^2(x)

The Pythagorean identity in trigonometry is a fundamental mathematical relationship that applies to right triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In terms of trigonometric functions, the Pythagorean identity is often written as:

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

However, you mentioned the Pythagorean identity equal to csc^2(x). Note that csc(x) is the reciprocal of the sin(x) function, which means csc(x) = 1/sin(x).

To express the Pythagorean identity in terms of csc(x), we can replace sin^2(x) with (1/csc^2(x)):

(1/csc^2(x)) + cos^2(x) = 1

To simplify this expression further, we can multiply every term by csc^2(x) to eliminate the fraction:

1 + csc^2(x) * cos^2(x) = csc^2(x)

Now, we can subtract csc^2(x) from both sides:

1 = csc^2(x) – csc^2(x) * cos^2(x)

Simplifying the right side of the equation:

1 = csc^2(x) * (1 – cos^2(x))

We can recognize that (1 – cos^2(x)) is equal to sin^2(x) by applying another trigonometric identity (specifically, the identity sin^2(x) + cos^2(x) = 1). Substituting sin^2(x) for (1 – cos^2(x)):

1 = csc^2(x) * sin^2(x)

Thus, the Pythagorean identity in terms of csc^2(x) is:

csc^2(x) * sin^2(x) = 1

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