The value of the expression `cos^3theta+cos^3(120+theta)+cos^3(240+theta)` is;

Options

A. `cos 3 theta`
B. `1/4 cos 3 theta`
C. `3/4 cos 3 theta`
D. None of these

Answer

To solve the expression \( \cos^3 \theta + \cos^3(120^\circ + \theta) + \cos^3(240^\circ + \theta) \), we can use the identity for the sum of cubes and properties of cosine.

### Step-by-step Solution:

1. **Identify the expression**:
\[ E = \cos^3 \theta + \cos^3(120^\circ + \theta) + \cos^3(240^\circ + \theta) \]

2. **Use the identity for the sum of cubes**:
The identity states that if \( a + b + c = 0 \), then:
\[ a^3 + b^3 + c^3 = 3abc \]
Here, we will let:
\[ a = \cos \theta, \quad b = \cos(120^\circ + \theta), \quad c = \cos(240^\circ + \theta) \]

3. **Check if \( a + b + c = 0 \)**:
We know that:
\[ \cos(120^\circ + \theta) = -\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]
\[ \cos(240^\circ + \theta) = -\frac{1}{2} \cos \theta + \frac{\sqrt{3}}{2} \sin \theta \]
Adding these:
\[ \cos \theta + \left(-\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta\right) + \left(-\frac{1}{2} \cos \theta + \frac{\sqrt{3}}{2} \sin \theta\right) = 0 \]
Thus, \( a + b + c = 0 \).

4. **Apply the sum of cubes identity**:
Since \( a + b + c = 0 \):
\[ E = 3abc \]

5. **Calculate \( abc \)**:
\[ abc = \cos \theta \cdot \cos(120^\circ + \theta) \cdot \cos(240^\circ + \theta) \]
Using the product:
\[ \cos(120^\circ + \theta) = -\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]
\[ \cos(240^\circ + \theta) = -\frac{1}{2} \cos \theta + \frac{\sqrt{3}}{2} \sin \theta \]
The product simplifies to:
\[ abc = \cos \theta \left(-\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta\right) \left(-\frac{1}{2} \cos \theta + \frac{\sqrt{3}}{2} \sin \theta\right) \]
This results in:
\[ = \cos \theta \left(\frac{1}{4} \cos^2 \theta - \frac{3}{4} \sin^2 \theta\right) \]

6. **Final expression**:
Therefore:
\[ E = 3 \left(\cos \theta \left(\frac{1}{4} \cos^2 \theta - \frac{3}{4} \sin^2 \theta\right)\right) \]
Simplifying gives:
\[ E = \frac{3}{4} \cos \theta ( \cos^2 \theta - 3 \sin^2 \theta ) \]
Using \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ E = \frac{3}{4} \cos \theta ( 4 \cos^2 \theta - 3 ) \]

7. **Conclusion**:
The final value of the expression is:
\[ E = \frac{3}{4} \cos 3\theta \]