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

Answer 1

Starting with the Pythagorean identity, we have:

1 + cot^2(θ) = csc^2(θ)

We know that cot(θ) = 1/tan(θ) and csc(θ) = 1/sin(θ), so we can rewrite the identity as:

1 + (1/tan(θ))^2 = (1/sin(θ))^2

Next, we simplify each side of the equation:

1 + 1/(tan(θ))^2 = 1/(sin(θ))^2

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

1 + 1/[(sin(θ))^2 * (cos(θ))^2] = 1/(sin(θ))^2

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

We know that sin^2(θ) + cos^2(θ) = 1, so we can rewrite the equation as:

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

This simplifies to:

1 = 1

Therefore, the Pythagorean identity is verified.