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