Verify that sec(θ)/csc(θ)-cot(θ) - sec(θ)/csc(θ)-cot(θ) = 2csc(θ) is an identity.

Question

can some help me through this? thank you!

Answer 1

sec(θ)/(csc(θ)-cot(θ)) - sec(θ)/(csc(θ)+cot(θ))
becomes

sec(θ)(csc(θ)+cot(θ)) - sec(θ)(csc(θ)-cot(θ))
----------------------------------------------------------
(csc(θ)-cot(θ))(csc(θ)+cot(θ))

Now expand the numerator and note that the denominator is just 1.