If `costheta + sec theta =2`, then the value of `cos^6theta + sec^6 theta` is

Options

A. 1
B. 2
C. 4
D. 8

Answer

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ \cos \theta + \sec \theta = 2 \]

2. **Substituting Secant**:
Recall that \(\sec \theta = \frac{1}{\cos \theta}\). We can rewrite the equation as:
\[ \cos \theta + \frac{1}{\cos \theta} = 2 \]

3. **Multiplying through by \(\cos \theta\)**:
To eliminate the fraction, multiply both sides by \(\cos \theta\):
\[ \cos^2 \theta + 1 = 2 \cos \theta \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ \cos^2 \theta - 2 \cos \theta + 1 = 0 \]

5. **Factoring the Quadratic**:
This can be factored as:
\[ (\cos \theta - 1)^2 = 0 \]

6. **Finding Cosine Value**:
Setting the factor equal to zero gives:
\[ \cos \theta - 1 = 0 \implies \cos \theta = 1 \]

7. **Finding Secant Value**:
Since \(\sec \theta = \frac{1}{\cos \theta}\), we have:
\[ \sec \theta = \frac{1}{1} = 1 \]

8. **Calculating \(\cos^6 \theta + \sec^6 \theta\)**:
Now we can find:
\[ \cos^6 \theta + \sec^6 \theta = 1^6 + 1^6 = 1 + 1 = 2 \]

Thus, the final answer is:
\[ \cos^6 \theta + \sec^6 \theta = 2 \]