If `costheta=(1)/(2)`,then find the value of `4cos^(3)theta-3costheta`.

Answer

To solve the problem where \( \cos \theta = \frac{1}{2} \) and we need to find the value of \( 4 \cos^3 \theta - 3 \cos \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Substitute the value of \( \cos \theta \)**:
We know that \( \cos \theta = \frac{1}{2} \). We will substitute this value into the expression \( 4 \cos^3 \theta - 3 \cos \theta \).

\[ 4 \cos^3 \theta - 3 \cos \theta = 4 \left( \frac{1}{2} \right)^3 - 3 \left( \frac{1}{2} \right) \]

2. **Calculate \( \left( \frac{1}{2} \right)^3 \)**:
Now, we calculate \( \left( \frac{1}{2} \right)^3 \).

\[ \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]

3. **Substitute back into the expression**:
Now substitute \( \frac{1}{8} \) back into the expression.

\[ 4 \left( \frac{1}{8} \right) - 3 \left( \frac{1}{2} \right) \]

4. **Calculate \( 4 \times \frac{1}{8} \)**:
Now calculate \( 4 \times \frac{1}{8} \).

\[ 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

5. **Calculate \( 3 \times \frac{1}{2} \)**:
Now calculate \( 3 \times \frac{1}{2} \).

\[ 3 \times \frac{1}{2} = \frac{3}{2} \]

6. **Combine the results**:
Now we combine the results from steps 4 and 5.

\[ \frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1 \]

### Final Answer:
Thus, the value of \( 4 \cos^3 \theta - 3 \cos \theta \) is \( -1 \).

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