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)
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