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

please help! thank you!

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.

To verify the given statement that sec(θ)/csc(θ) - cot(θ) - sec(θ)/csc(θ) - cot(θ) = 2csc(θ) is an identity, we need to simplify both sides of the equation and show that they are equal.

Let's start by simplifying the left side of the equation step by step:

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

Now, to simplify sec(θ)/csc(θ), we can use the reciprocal identities:

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

Therefore, the expression becomes:

(sin^2(θ) / cos(θ)) - cot(θ) - (sin^2(θ) / cos(θ)) - cot(θ)

Now, let's simplify cot(θ) as well:

cot(θ) = 1/tan(θ)
= cos(θ) / sin(θ)

Substituting this back into the expression:

(sin^2(θ) / cos(θ)) - (cos(θ) / sin(θ)) - (sin^2(θ) / cos(θ)) - (cos(θ) / sin(θ))

Next, we can combine like terms:

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

The sin^2(θ) terms cancel out:

0 / cos(θ) - (2cos(θ)) / sin(θ)

0 - (2cos(θ)) / sin(θ)

= -2cos(θ) / sin(θ)

Finally, we simplify the right side of the equation:

2csc(θ)

Using the reciprocal identities, csc(θ) = 1/sin(θ):

2 * (1/sin(θ))

= 2/sin(θ)

Now we can see that the left side of the equation, -2cos(θ)/sin(θ), is equal to the right side of the equation, 2/sin(θ).

Thus, we have verified that sec(θ)/csc(θ) - cot(θ) - sec(θ)/csc(θ) - cot(θ) = 2csc(θ) is an identity.