Answer :
Answer:
To find the value of the given expression:
\[3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6x + \cos^6x).\]
Let's simplify each term step by step:
1. For \(3(\sin x - \cos x)^4\):
\[= 3(\sin^4x - 4\sin^3x\cos x + 6\sin^2x\cos^2x - 4\sin x \cos^3x + \cos^4x).\]
2. For \(6(\sin x + \cos x)^2\):
\[= 6(\sin^2x + 2\sin x \cos x + \cos^2x).\]
3. For \(4(\sin^6x + \cos^6x)\):
\[= 4((\sin^2x)^3 + (\cos^2x)^3).\]
Now, let's simplify each term individually:
1. \(3(\sin x - \cos x)^4 = 3(\sin^4x - 4\sin^3x\cos x + 6\sin^2x\cos^2x - 4\sin x \cos^3x + \cos^4x).\)
\[= 3(\sin^4x - 4\sin^3x\cos x + 6\sin^2x\cos^2x - 4\sin x \cos^3x + \cos^4x).\]
2. \(6(\sin x + \cos x)^2 = 6(\sin^2x + 2\sin x \cos x + \cos^2x).\)
\[= 6(\sin^2x + 2\sin x \cos x + \cos^2x).\]
3. \(4(\sin^6x + \cos^6x) = 4((\sin^2x)^3 + (\cos^2x)^3).\)
\[= 4(\sin^6x + \cos^6x).\]
Now, let's simplify further and combine like terms:
\[3(\sin^4x - 4\sin^3x\cos x + 6\sin^2x\cos^2x - 4\sin x \cos^3x + \cos^4x) + 6(\sin^2x + 2\sin x \cos x + \cos^2x) + 4(\sin^6x + \cos^6x).\]
\[= 3\sin^4x - 12\sin^3x\cos x + 18\sin^2x\cos^2x - 12\sin x \cos^3x + 3\cos^4x + 6\sin^2x + 12\sin x \cos x + 6\cos^2x + 4\sin^6x + 4\cos^6x.\]
Now, let's group similar terms:
\[= (3\sin^4x + 6\sin^2x + 4\sin^6x) + (- 12\sin^3x\cos x - 12\sin x \cos^3x + 12\sin x \cos x) + (18\sin^2x\cos^2x + 6\cos^2x + 3\cos^4x + 4\cos^6x).\]
\[= \sin^2x(3\sin^2x + 6 + 4\sin^4x) - 12\sin x \cos x(\sin^2x + \cos^2x - 1) + \cos^2x(18\sin^2x + 6 + 3\cos^2x + 4\cos^4x).\]
Now, we'll use the identities \(\sin^2x + \cos^2x = 1\) and \(\sin^2x + \cos^2x = 1\) to simplify:
\[= \sin^2x(3\sin^2x + 6 + 4(1 - \cos^2x)) - 12\sin x \cos x(1 - 1) + \cos^2x(18\sin^2x + 6 + 3(1 - \sin^2x) + 4(1 - \sin^2x)^2).\]
\[= \sin^2x(3\sin^2x + 10 - 4\cos^2x) + \cos^2x(18\sin^2x + 9 - 3\sin^2x + 4(1 - 2\sin^2x + \sin^4x)).\]
\[= 3\sin^4x + 10\sin^2x - 4\sin^2x\cos^2x + 18\sin^2x\cos^2x + 9\cos^2x - 3\sin^2x\cos^2x + 4 - 8\sin^2x + 4\sin^4x).\]
\[= 7\sin^4x - 11\sin^2x + 14\sin^2x\cos^2x + 9\cos^2x + 4.\]
\[= 7\sin^4x - 11\sin^2x + 14\sin^2x(1 - \sin^2x) + 9(1 - \sin^2x) + 4.\]
\[= 7\sin^4x - 11\sin^2x + 14\sin^2x - 14\sin^4x + 9 - 9\sin^2x + 4.\]
\[= -7\sin^4x + 3\sin^2x + 13.\]
So, the value of the expression is \(-7\sin^4x + 3\sin^2x + 13\).
Answer:
To find the value of the expression \(3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x)\), let's simplify each term individually:
1. \(3(\sin x - \cos x)^4\):
\[= 3(\sin^4 x - 4\sin^3 x \cos x + 6\sin^2 x \cos^2 x - 4\sin x \cos^3 x + \cos^4 x)\]
2. \(6(\sin x + \cos x)^2\):
\[= 6(\sin^2 x + 2\sin x \cos x + \cos^2 x)\]
\[= 6(1 + 2\sin x \cos x)\] (since \(\sin^2 x + \cos^2 x = 1\))
3. \(4(\sin^6 x + \cos^6 x)\):
\[= 4(\sin^6 x + \cos^6 x)\]
Now, let's add up all the simplified terms:
\[3(\sin^4 x - 4\sin^3 x \cos x + 6\sin^2 x \cos^2 x - 4\sin x \cos^3 x + \cos^4 x) + 6(1 + 2\sin x \cos x) + 4(\sin^6 x + \cos^6 x)\]
\[= 3\sin^4 x - 12\sin^3 x \cos x + 18\sin^2 x \cos^2 x - 12\sin x \cos^3 x + 3\cos^4 x + 6 + 12\sin x \cos x + 4(\sin^6 x + \cos^6 x)\]
Now, observe that \( \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \).
So, we can rewrite the expression as:
\[= 3(1 - 2\sin^2 x \cos^2 x) - 12\sin^3 x \cos x + 18\sin^2 x \cos^2 x - 12\sin x \cos^3 x + 6 + 12\sin x \cos x + 4(\sin^6 x + \cos^6 x)\]
\[= 3 - 6\sin^2 x \cos^2 x - 12\sin^3 x \cos x + 18\sin^2 x \cos^2 x - 12\sin x \cos^3 x + 6 + 12\sin x \cos x + 4(\sin^6 x + \cos^6 x)\]
\[= 9 + 6\sin^2 x \cos^2 x - 12\sin^3 x \cos x - 12\sin x \cos^3 x + 4(\sin^6 x + \cos^6 x)\]
Now, we need to simplify \(4(\sin^6 x + \cos^6 x)\) using the identity \( \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)^3 - 3\sin^2 x \cos^2 x (\sin^2 x + \cos^2 x) = 1 - 3\sin^2 x \cos^2 x \).
So, the expression becomes:
\[9 + 6\sin^2 x \cos^2 x - 12\sin^3 x \cos x - 12\sin x \cos^3 x + 4(1 - 3\sin^2 x \cos^2 x)\]
\[= 9 + 6\sin^2 x \cos^2 x - 12\sin^3 x \cos x - 12\sin x \cos^3 x + 4 - 12\sin^2 x \cos^2 x\]
\[= 13 - 6\sin^2 x \cos^2 x - 12\sin^3 x \cos x - 12\sin x \cos^3 x\]
\[= \boxed{13 - 6\sin^2 x \cos^2 x - 12\sin x \cos x (\sin x + \cos x)}\]
This is the simplified expression for \(3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x)\).