verify the Pythagorean identity. 1+cot^2θ=csc^2θ

Answer 1

Starting with the left side of the identity:

1 + cot^2θ

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

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

= (2sin^2θ + cos^2θ) / cos^2θ

= (2sin^2θ / cos^2θ) + 1

= 2(csc^2θ - 1) + 1 (using the reciprocal identities)

= 2csc^2θ - 2 + 1

= 2csc^2θ - 1

Which is equal to the right side of the identity, therefore the Pythagorean identity is verified.

Answer 2

what a lot of work!
Start with the most basic of trig identities:
sin^2θ + cos^2θ = 1
and then just divide by sin^2θ to get
1 + cot^2θ = csc^2θ

Answer 3

That's a much simpler approach, and it's a more direct way to arrive at the Pythagorean identity.

Starting with the basic identity sin^2θ + cos^2θ = 1:

Add cot^2θ to both sides:

sin^2θ + cos^2θ + cot^2θ = 1 + cot^2θ

Recall that cot^2θ = cos^2θ / sin^2θ:

sin^2θ + cos^2θ + cos^2θ / sin^2θ = csc^2θ

Combine the terms with a common denominator:

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

Simplify using the identity sin^2θ + cos^2θ = 1:

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

Distribute and simplify:

(sin^2θ - sin^2θ * cos^2θ + cos^2θ) / sin^2θ = csc^2θ

Simplify further by factoring out sin^2θ:

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

Simplify and recognize the expression on the left side of the equation:

1 / sin^2θ = csc^2θ

Rearrange to get the Pythagorean identity:

1 + cot^2θ = csc^2θ

This approach is much quicker and requires less algebraic manipulation compared to the previous method.