Prove the following identitie is true (18)?

Question

((sin^4 t-cos^4 t)/(sin^2 t cos^2t))= sec^2 t - csc^2 t

Answer 1

(sin^4 t-cos^4 t)/(sin^2 t cos^2t)
= (sin^2 t + cos^2 t) (sin^2 t - cos^2 t) / (sin^2 t cos^2 t)
= (1)(sin^2 t - cos^2 t) / (sin^2 t cos^2 t)
= sin^2 t - cos^2 t / (sin^2 t cos^2 t)
= sin^2 t / (sin^2 t cos^2 t) - cos^2 t / (sin^2 t cos^2 t)
= 1 / cos^2 t - 1 / sin^2 t
= sec^2 t - csc^2 t

Answer 2

sin^4 t - cos^4 t = (sin^2 t - cos^2 t)(sin^2 t+ cos^2 t) = sin^2 t- cos^2 t as sin^2 t+ cos^2 t =1

so LHS = (sin^2 t- cos^2 t)/ sin^2 t cos ^2 t

= 1/ cos^2 t - 1/sin ^2 t = sec^2 t- csc^2 t

Answer 3

Prove the identity.

(sin^4 t - cos^4 t) / (sin²t cos²t) = sec²t - csc²t

Let's work with the left hand side.

Left Hand Side = (sin^4 t - cos^4 t) / (sin²t cos²t)

(sin²t + cos²t)(sin²t - cos²t) / (sin²t cos²t)

= (1)(sin²t - cos²t) / (sin²t cos²t)

= sin²t / (sin²t cos²t) - cos²t / (sin²t cos²t)

= 1 / cos²t - 1 / sin²t = sec²t - csc²t = Right Hand Side

Answer 4

csc x = one million/sin x cot x = one million/tan x csc^2x = one million/sin^2x cot^2x = one million/tan^2x (one million/sin^2x) - (one million/tan^2x) = one million (one million/sin^2x) - (one million/(sin^2x/cos^2x)) = one million (one million/sin^2x) - (cos^2x/sin^2x) = one million (one million-cos^2x)/(sin^2x) = one million (sin^2x/sin^2x) = one million one million = one million