How do you factor and simplify,

1-sin^3x

and
sec^4x-2sec^2xtan^2x+tan^4x

the first is the difference of two cubes:

(1-a^3)=(1-a)(1+a+a^2)
from...
a3 – b3 = (a – b)(a2 + ab + b2)

the second is very similar to

(a^2-2ab+b^2).....

I'm not positive, but the answer to 1-sin^3 would be (1-sinx)(1+sinx+sin^2x)

To factor and simplify expressions, we first need to identify any common factors and then apply algebraic techniques to simplify the expressions further. Let's break down each expression step-by-step.

1. Simplifying 1 - sin^3(x):

Step 1: Recognize the pattern
We notice that the given expression is in the form of a difference of cubes. A difference of cubes can be factored using the formula:
a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Step 2: Apply the formula
Using the formula for a difference of cubes, we have:
1 - sin^3(x) = (1 - sin(x))(1 + sin(x) + sin^2(x)).

This is the factored and simplified form of the expression 1 - sin^3(x).

2. Simplifying sec^4(x) - 2sec^2(x)tan^2(x) + tan^4(x):

Step 1: Recognize the pattern
We observe that the given expression is in the form of two perfect squares: (sec^2(x))^2 and (tan^2(x))^2.

Step 2: Substitute trigonometric identities
Using the Pythagorean identity for sec^2(x), we know that sec^2(x) = 1 + tan^2(x). Substituting this identity, we get:
sec^4(x) - 2sec^2(x)tan^2(x) + tan^4(x) = (1 + tan^2(x))^2 - 2(1 + tan^2(x))tan^2(x) + tan^4(x).

Step 3: Simplify the expression
Expanding the squared term using the rule (a + b)^2 = a^2 + 2ab + b^2, we have:
(1 + tan^2(x))^2 - 2(1 + tan^2(x))tan^2(x) + tan^4(x) = 1 + 2tan^2(x) + tan^4(x) - 2tan^2(x) - 2tan^4(x) + tan^4(x).

After simplifying like terms, the final result is:
1 - tan^2(x).

This is the simplified form of the expression sec^4(x) - 2sec^2(x)tan^2(x) + tan^4(x).