HELP!!!! I don't know how to do the trig identity with this problem

csc^4 x-cot^4x= Csx^2 x + cot^2x

recall that a^2 - b^2 = (a+b)(a-b)

csc^4 - cot^4 = (csc^2 - cot^2)(csc^2 + cot^2)

if you recall that csc^2 = 1 + cot^2 the rest is downhill. . .

No problem! I'd be happy to help you with this trig identity problem.

The identity you are dealing with is:
csc^4(x) - cot^4(x) = csc^2(x) + cot^2(x)

To solve this, we can use some of the properties and identities of trigonometric functions.

Step 1: Write the left-hand side (LHS) in terms of sine and cosine.
Recall that csc(x) is the reciprocal of sin(x), and cot(x) is the reciprocal of tan(x). We can rewrite csc^4(x) and cot^4(x) using these definitions:
csc^4(x) = (1/sin(x))^4 = 1/sin^4(x)
cot^4(x) = (1/tan(x))^4 = (cos^2(x))/(sin^2(x))^2

Step 2: Write the right-hand side (RHS) in terms of sine and cosine.
Recall that csc^2(x) = 1/sin^2(x) and cot^2(x) = (cos^2(x))/(sin^2(x)).

Step 3: Substitute the expressions from Step 1 and Step 2 into the original equation.
We have:
(1/sin^4(x)) - ((cos^2(x))/(sin^2(x))^2) = 1/sin^2(x) + (cos^2(x))/(sin^2(x))

Step 4: Simplify the equation.
To simplify, we need to find a common denominator for the fractions on both sides of the equation. The common denominator is sin^4(x). Multiply each term by sin^4(x):

1 - cos^4(x) = sin^2(x) + (cos^2(x))(sin^4(x))

Step 5: Use the Pythagorean identity.
The Pythagorean identity states that sin^2(x) + cos^2(x) = 1. We can rewrite the equation using this identity:

1 - cos^4(x) = sin^2(x) + (cos^2(x))(1 - cos^2(x))

Step 6: Simplify the equation further.
Expand the term on the right side:
1 - cos^4(x) = sin^2(x) + cos^2(x) - (cos^2(x))(cos^2(x))

Simplify the equation by combining like terms:
1 - cos^4(x) = sin^2(x) + cos^2(x) - cos^4(x)

Step 7: Cancel out common terms.
Note that -cos^4(x) and cos^4(x) cancel out on both sides of the equation. We are left with:
1 = sin^2(x) + cos^2(x)

Step 8: Recognize the final equation.
The equation 1 = sin^2(x) + cos^2(x) is one of the fundamental trigonometric identities, known as the Pythagorean identity. It shows that the square of the sine plus the square of the cosine of any angle x is equal to 1.

So, the original identity csc^4(x) - cot^4(x) = csc^2(x) + cot^2(x) simplifies to the Pythagorean identity 1 = sin^2(x) + cos^2(x).

I hope this explanation helps you understand how to approach this type of trigonometry problem. Let me know if you have any further questions!