If `theta=20^(@)`, then `8cos^(3)theta-6 cos theta` is

Options

A. `-1`
B. 1
C. 0
D. 2

Answer

To solve the problem, we need to evaluate the expression \( 8 \cos^3 \theta - 6 \cos \theta \) for \( \theta = 20^\circ \).

### Step-by-Step Solution:

1. **Identify the expression**: We have \( 8 \cos^3 \theta - 6 \cos \theta \).

2. **Use the cosine identity**: We know from trigonometric identities that:
\[ \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \]
We can rearrange this identity to express \( 8 \cos^3 \theta - 6 \cos \theta \) in terms of \( \cos 3\theta \).

3. **Rearranging the identity**: Multiply the identity by 2:
\[ 2 \cos 3\theta = 2(4 \cos^3 \theta - 3 \cos \theta) = 8 \cos^3 \theta - 6 \cos \theta \]
Thus, we can rewrite our expression as:
\[ 8 \cos^3 \theta - 6 \cos \theta = 2 \cos 3\theta \]

4. **Substituting \( \theta \)**: Now substitute \( \theta = 20^\circ \):
\[ 8 \cos^3(20^\circ) - 6 \cos(20^\circ) = 2 \cos(3 \times 20^\circ) = 2 \cos(60^\circ) \]

5. **Calculating \( \cos(60^\circ) \)**: We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]

6. **Final Calculation**: Now substitute \( \cos(60^\circ) \) back into the expression:
\[ 2 \cos(60^\circ) = 2 \times \frac{1}{2} = 1 \]

### Final Answer:
Thus, the value of \( 8 \cos^3(20^\circ) - 6 \cos(20^\circ) \) is \( 1 \).

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