Prove that :`cos ((pi)/(3) + x) = ( cos x - sqrt3 sin x )/(2)`

Answer

To prove that \( \cos\left(\frac{\pi}{3} + x\right) = \frac{\cos x - \sqrt{3} \sin x}{2} \), we will start with the left-hand side (LHS) and use the cosine addition formula.

### Step-by-Step Solution:

1. **Start with the LHS**:
\[ \cos\left(\frac{\pi}{3} + x\right) \]

**Hint**: Remember that we will use the cosine addition formula.

2. **Apply the Cosine Addition Formula**:
The cosine addition formula states that:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
Here, let \( a = \frac{\pi}{3} \) and \( b = x \). Therefore, we have:
\[ \cos\left(\frac{\pi}{3} + x\right) = \cos\left(\frac{\pi}{3}\right) \cos x - \sin\left(\frac{\pi}{3}\right) \sin x \]

**Hint**: Identify the values of \( \cos\left(\frac{\pi}{3}\right) \) and \( \sin\left(\frac{\pi}{3}\right) \).

3. **Substitute the Values**:
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \quad \text{and} \quad \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Substituting these values into the equation gives:
\[ \cos\left(\frac{\pi}{3} + x\right) = \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \]

**Hint**: Factor out the common denominator.

4. **Factor Out the Denominator**:
We can factor out \( \frac{1}{2} \) from the right-hand side:
\[ \cos\left(\frac{\pi}{3} + x\right) = \frac{1}{2} \left( \cos x - \sqrt{3} \sin x \right) \]

**Hint**: Rewrite this to match the right-hand side (RHS).

5. **Final Form**:
This can be rewritten as:
\[ \cos\left(\frac{\pi}{3} + x\right) = \frac{\cos x - \sqrt{3} \sin x}{2} \]

**Hint**: Compare this with the RHS.

6. **Conclusion**:
Since we have shown that the LHS equals the RHS, we conclude that:
\[ \cos\left(\frac{\pi}{3} + x\right) = \frac{\cos x - \sqrt{3} \sin x}{2} \]
Hence, the statement is proved.