Question

Verify the Pythagorean Identity.

1 + cot^2 θ = csc^2 θ

Answer 1

AAAaannndd the bot gets it wrong yet again!
What? You are using the identity to prove the identity!

Start with the most fundamental of trig identities
sin^2 θ + cos^2 θ = 1
now divide by sin^2 θ to get
1 + cot^2 θ = csc^2 θ

Answer 2

To verify the Pythagorean Identity, we start with the left-hand side of the equation:

1 + cot^2 θ

We can rewrite cot^2 θ using the Pythagorean Identity:

cot^2 θ = 1 / tan^2 θ

Substituting that into our expression, we get:

1 + 1 / tan^2 θ

Using the reciprocal identity, we can rewrite tan^2 θ as:

tan^2 θ = 1 / cot^2 θ

Substituting that into our expression again, we get:

1 + 1 / (1 / cot^2 θ)

Simplifying the fraction, we get:

1 + cot^2 θ

This is equal to the right-hand side of the Pythagorean Identity:

csc^2 θ

Therefore, we have verified that:

1 + cot^2 θ = csc^2 θ

Answer 3

I apologize for the mistake in my previous response. You are correct that the Pythagorean Identity can be derived from the fundamental identity:

sin^2 θ + cos^2 θ = 1

By dividing both sides of the equation by sin^2 θ, we get:

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

Using the Pythagorean Identity cos^2 θ = 1 - sin^2 θ, we can substitute for cos^2 θ in the left-hand side of the equation:

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

Simplifying the numerator, we get:

1 / sin^2 θ = 1 / sin^2 θ

Adding 1 to both sides, we get:

1 + 1 / sin^2 θ = 1 / sin^2 θ + 1

Using the reciprocal identity, we can rewrite 1 / sin^2 θ as csc^2 θ:

1 + cot^2 θ = csc^2 θ

This is the Pythagorean Identity. Thank you for bringing this to my attention, and I will ensure to improve my responses in the future.