`2(sin^6theta+cos^6theta)-3(sin^4theta+cos^4theta)` is equal to `0` (b) `1` (c) `-1` (d) None of these

Options

A. 2
B. 0
C. 4
D. 6

Answer

To solve the expression \(2(\sin^6 \theta + \cos^6 \theta) - 3(\sin^4 \theta + \cos^4 \theta)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 2(\sin^6 \theta + \cos^6 \theta) - 3(\sin^4 \theta + \cos^4 \theta) \]

### Step 2: Use the identity for \(a^3 + b^3\)
Recall that \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, let \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\). Therefore, we can write:
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:
\[ \sin^6 \theta + \cos^6 \theta = \sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta \]

### Step 3: Substitute back into the expression
Now substituting this back into the original expression:
\[ 2(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) - 3(\sin^4 \theta + \cos^4 \theta) \]

### Step 4: Simplify the expression
Distributing the 2:
\[ 2\sin^4 \theta - 2\sin^2 \theta \cos^2 \theta + 2\cos^4 \theta - 3\sin^4 \theta - 3\cos^4 \theta \]
Combine like terms:
\[ (2\sin^4 \theta - 3\sin^4 \theta) + (2\cos^4 \theta - 3\cos^4 \theta) - 2\sin^2 \theta \cos^2 \theta \]
This simplifies to:
\[ -\sin^4 \theta - \cos^4 \theta - 2\sin^2 \theta \cos^2 \theta \]

### Step 5: Use the identity for \(a^4 + b^4\)
Recall that \(a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2\). Here, we have:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]

### Step 6: Substitute this back into the expression
Now substituting this back:
\[ -(1 - 2\sin^2 \theta \cos^2 \theta) - 2\sin^2 \theta \cos^2 \theta \]
This simplifies to:
\[ -1 + 2\sin^2 \theta \cos^2 \theta - 2\sin^2 \theta \cos^2 \theta \]
The \(2\sin^2 \theta \cos^2 \theta\) terms cancel out:
\[ -1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{-1} \]