Prove that the equation is an identity:

cot4theta - csc4theta = -cot sq theta - csc sq theta

How ??

cot^4 - csc^4

= (cot^2 - csc^2)(cot^2 + csc^2)
= (-1)(cot^2 + csc^2)
= -cot^2 - csc^2

remember csc^2 = 1 + cot^2 ??

To prove that the given equation is an identity, we need to show that it holds true for all values of theta. Here's how we can do it:

1. Start with the left side of the equation: cot^4(theta) - csc^4(theta)
2. Rewrite cot^4(theta) as (cot^2(theta))^2 and csc^4(theta) as (csc^2(theta))^2
3. Now, the equation becomes (cot^2(theta))^2 - (csc^2(theta))^2

Next, let's recall the Pythagorean identities:

cot^2(theta) + 1 = csc^2(theta) ---(1)
tan^2(theta) + 1 = sec^2(theta) ---(2)

4. Rearrange equation (1) to isolate cot^2(theta): cot^2(theta) = csc^2(theta) - 1
5. Substitute this into the equation from step 3:

(csc^2(theta) - 1)^2 - (csc^2(theta))^2
= (csc^2(theta))^2 - 2(csc^2(theta)) + 1 - (csc^2(theta))^2
= -2(csc^2(theta)) + 1

6. Simplify the right side by using the identity csc^2(theta) = 1 + cot^2(theta):

= -2(1 + cot^2(theta)) + 1
= -2 - 2(cot^2(theta)) + 1
= -2cot^2(theta) - 1

Now, we have the right side of the original equation: -cot^2(theta) - csc^2(theta)

7. Finally, compare the right side of the original equation with the right side we obtained after simplification. They are equal:

-2cot^2(theta) - 1 = -cot^2(theta) - csc^2(theta)

Therefore, the left side of the equation equals the right side for all values of theta, making it an identity.