To simplify and find the value of [tex]\( \sin^4 A - \cos^4 A - 2 \sin^2 A \)[/tex], follow these steps:
1. Rewrite the given expression:
[tex]\[
\sin^4 A - \cos^4 A - 2 \sin^2 A
\][/tex]
2. Factor the expression [tex]\( \sin^4 A - \cos^4 A \)[/tex]:
We can use the difference of squares formula:
[tex]\[
\sin^4 A - \cos^4 A = (\sin^2 A)^2 - (\cos^2 A)^2 = (\sin^2 A - \cos^2 A)(\sin^2 A + \cos^2 A)
\][/tex]
3. Use the Pythagorean identity:
[tex]\[
\sin^2 A + \cos^2 A = 1
\][/tex]
Substituting this into the expression, we get:
[tex]\[
(\sin^2 A - \cos^2 A)(\sin^2 A + \cos^2 A) = (\sin^2 A - \cos^2 A) \cdot 1 = \sin^2 A - \cos^2 A
\][/tex]
4. Substitute the simplified form back in:
Our expression now becomes:
[tex]\[
\sin^2 A - \cos^2 A - 2 \sin^2 A
\][/tex]
5. Combine like terms:
[tex]\[
\sin^2 A - \cos^2 A - 2 \sin^2 A = -\cos^2 A - \sin^2 A
\][/tex]
6. Use the Pythagorean identity again:
[tex]\[
\sin^2 A + \cos^2 A = 1
\][/tex]
Therefore,
[tex]\[
- \cos^2 A - \sin^2 A = -1
\][/tex]
Thus, the simplified value of [tex]\( \sin^4 A - \cos^4 A - 2 \sin^2 A \)[/tex] is:
[tex]\[
\boxed{-1}
\][/tex]
