MATH SOLVE

4 months ago

Q:
# I really need help with these questions:Use basic identities to find the simplified expression-1. (cos^2 x + sin^2 x) / (cot^2 x - csc^2 x)2. cosine θ^2 / sine θ^2 + csc θ sin θExplanations would be greatly appreciated

Accepted Solution

A:

If we take the Pythagorean identity identity sin^2 x + cos^2 x = 1 then

(cos^2 x + sin^2 x) / (cot^2 x - csc^2 x)

The numerator becomes 1 since addition order matters not.

1 / (cot^2 x - csc^2 x)

If we factor the denominator out a negative

1 / -(csc^2 x - cot^2 x)

Consider sin^2 x + cos^2 x = 1. Divide both sides by sin^2 x to get

1 + cot^2 x = csc^2 x

Subtract both sides by cot^2 x to get 1 = csc^2 x - cot^2 x.

Replace the denominator

1 / -(1) = -1

For cos^2 θ / sin^2 θ + csc θ sin θ, we use cscθ = 1/sinθ and cosθ/sinθ = cotθ so

= cos^2 θ / sin^2 θ + 1

= cot^2 θ + 1

We use 1 + cot^2 θ = csc^2 θ to simplify this to

= csc^2 θ

Answers: -1

csc^2 θ

(cos^2 x + sin^2 x) / (cot^2 x - csc^2 x)

The numerator becomes 1 since addition order matters not.

1 / (cot^2 x - csc^2 x)

If we factor the denominator out a negative

1 / -(csc^2 x - cot^2 x)

Consider sin^2 x + cos^2 x = 1. Divide both sides by sin^2 x to get

1 + cot^2 x = csc^2 x

Subtract both sides by cot^2 x to get 1 = csc^2 x - cot^2 x.

Replace the denominator

1 / -(1) = -1

For cos^2 θ / sin^2 θ + csc θ sin θ, we use cscθ = 1/sinθ and cosθ/sinθ = cotθ so

= cos^2 θ / sin^2 θ + 1

= cot^2 θ + 1

We use 1 + cot^2 θ = csc^2 θ to simplify this to

= csc^2 θ

Answers: -1

csc^2 θ