Verify the Pythagorean Identity.1+cot2θ=csc2θ show all work in short term

Answer 1

To verify the Pythagorean identity 1 + cot^2(θ) = csc^2(θ), we start with the definition of cotangent and cosecant:

cot(θ) = cos(θ) / sin(θ) and csc(θ) = 1 / sin(θ)

Now, we square both sides of the cotangent formula:

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

Next, we substitute back into the Pythagorean identity:

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

Now, we simplify the right side using trigonometric identities:

1 + cos^2(θ) / sin^2(θ) = sin^2(θ) / sin^2(θ) + cos^2(θ) / sin^2(θ)
= (sin^2(θ) + cos^2(θ)) / sin^2(θ)
= 1 / sin^2(θ)
= csc^2(θ)

Therefore, we have verified the Pythagorean identity 1 + cot^2(θ) = csc^2(θ).