Proving the Pythagorean Identity: A Step-by-Step Guide

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

To prove the Pythagorean Identity, we start with the definition of cotangent:

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

We can square both sides of this equation:

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

Using the definition of tangent:

tan(x) = sin(x) / cos(x)

We can rewrite the right-hand side of the equation as:

cos^2(x) / sin^2(x) = (1 / sin^2(x)) * cos^2(x)

Now, let’s focus on the left-hand side of the equation:

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

We can rewrite the right-hand side of the equation as:

(1 / sin^2(x)) * cos^2(x) = cos^2(x) / sin^2(x)

Now, we can compare the left-hand side and the right-hand side of the equation:

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

We have now established that the left-hand side is equal to the right-hand side of the equation

To prove the Pythagorean Identity, we start with the definition of cotangent:

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

We can square both sides of this equation:

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

Using the definition of tangent:

tan(x) = sin(x) / cos(x)

We can rewrite the right-hand side of the equation as:

cos^2(x) / sin^2(x) = (1 / sin^2(x)) * cos^2(x)

Now, let’s focus on the left-hand side of the equation:

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

We can rewrite the right-hand side of the equation as:

(1 / sin^2(x)) * cos^2(x) = cos^2(x) / sin^2(x)

Now, we can compare the left-hand side and the right-hand side of the equation:

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

We have now established that the left-hand side is equal to the right-hand side of the equation. Therefore, we can conclude that:

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

Using the identity cos^2(x) + sin^2(x) = 1, we can further simplify:

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

Finally, we have proven the Pythagorean Identity:

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

More Answers: