Verify the Pythagorean Identity.
1 + cot^2 θ = csc^2 θ
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
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.
To verify the Pythagorean Identity 1 + cot^2 θ = csc^2 θ, we need to manipulate the expressions on both sides of the equation to see if they are equivalent.
Let's start with the left side of the equation:
1 + cot^2 θ
Now, we can rewrite cot^2 θ as:
1 + (cos θ / sin θ)^2
To simplify this expression, we can square the entire term by multiplying the numerator and denominator by sin^2 θ:
(1 * sin^2 θ + cos^2 θ) / sin^2 θ
Using the Pythagorean Identity sin^2 θ + cos^2 θ = 1, we can replace sin^2 θ + cos^2 θ in the numerator:
1 / sin^2 θ
Since csc θ is the reciprocal of sin θ, we can rewrite this expression as:
csc^2 θ
Therefore, we have:
1 + cot^2 θ = csc^2 θ
This verifies the Pythagorean Identity.
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