The value of the expression `sin^6 theta + cos^6 theta + 3 sin^2 theta. cos^2 theta` equals

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the expression \( \sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta \), we can use the identity for the sum of cubes and some trigonometric identities. Here’s a step-by-step solution:

### Step 1: Recognize the Structure
We can recognize that the expression \( \sin^6 \theta + \cos^6 \theta \) can be rewritten using the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
where \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \).

### Step 2: Apply the Identity
Using the identity, we rewrite:
\[ \sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ \sin^6 \theta + \cos^6 \theta = 1 \cdot (\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta) = \sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta \]

### Step 3: Simplify Further
Next, we can simplify \( \sin^4 \theta + \cos^4 \theta \) using the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta \]
Substituting \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^4 \theta + \cos^4 \theta = 1^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]

### Step 4: Combine Everything
Now, substituting back into our expression:
\[ \sin^6 \theta + \cos^6 \theta = (1 - 2\sin^2 \theta \cos^2 \theta) - \sin^2 \theta \cos^2 \theta \]
This simplifies to:
\[ \sin^6 \theta + \cos^6 \theta = 1 - 3\sin^2 \theta \cos^2 \theta \]

### Step 5: Substitute Back into the Original Expression
Now, substituting this back into the original expression:
\[ \sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta = (1 - 3\sin^2 \theta \cos^2 \theta) + 3 \sin^2 \theta \cos^2 \theta \]
This simplifies to:
\[ 1 - 3\sin^2 \theta \cos^2 \theta + 3\sin^2 \theta \cos^2 \theta = 1 \]

### Final Answer
Thus, the value of the expression \( \sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta \) is:
\[ \boxed{1} \]