To determine whether the equation [tex]\(\cos^6(A) + \sin^6(A) = 1 - 3 \sin^2(A) \cos^2(A)\)[/tex] is true, we need to simplify both sides of the equation and check if they are equal.
### Step-by-Step Solution:
1. Simplify the Left-Hand Side (LHS):
[tex]\[
\cos^6(A) + \sin^6(A)
\][/tex]
2. Expression for [tex]\( \cos^6(A) + \sin^6(A) \)[/tex]:
We can use the identity for sums of cubes:
[tex]\[
x^6 + y^6 = (x^2 + y^2)(x^4 - x^2y^2 + y^4)
\][/tex]
Let [tex]\( x = \cos(A) \)[/tex] and [tex]\( y = \sin(A) \)[/tex]. Since we know:
[tex]\[
\cos^2(A) + \sin^2(A) = 1
\][/tex]
We can now rewrite the expression:
[tex]\[
\cos^6(A) + \sin^6(A) = 1^3 - 3 \cos^2(A) \sin^2(A) (1)
\][/tex]
Simplifying this expression, we get:
[tex]\[
\cos^6(A) + \sin^6(A) = 1 - 3 \cos^2(A) \sin^2(A)
\][/tex]
3. Compare with Right-Hand Side (RHS):
[tex]\[
1 - 3 \sin^2(A) \cos^2(A)
\][/tex]
4. Verification:
Let's compare the simplified left-hand side with the right-hand side. After simplifying both sides, we obtain:
[tex]\[
\sin^6(A) + \cos^6(A) = \frac{3 \cos(4A)}{8} + \frac{5}{8}
\][/tex]
And the right-hand side is:
[tex]\[
1 - 3 \sin^2(A) \cos^2(A) \neq \sin^6(A) + \cos^6(A)
\][/tex]
### Conclusion:
From our calculations, we see that:
[tex]\[
\cos^6(A) + \sin^6(A) \neq 1 - 3 \sin^2(A) \cos^2(A)
\][/tex]
Thus, the given equation:
[tex]\[
\cos^6(A) + \sin^6(A) = 1 - 3 \sin^2(A) \cos^2(A)
\][/tex]
is not true.
