Question

For an acute angle $\theta$, if $\cos \theta = \frac{1}{8}$, then $\frac{8\sec\theta + 1}{8\sec\theta - 1}$ equals

• A. $\frac{64}{63}$
• B. $0$
• C. $\frac{65}{63}$
• D. $1$

Hint:

Converting secant to cosine inside the expression can also simplify the calculation directly:
\[ \frac{\frac{8}{\cos\theta} + 1}{\frac{8}{\cos\theta} - 1} = \frac{8 + \cos\theta}{8 - \cos\theta} \]
Substituting $\cos\theta = \frac{1}{8}$ gives $\frac{8 + 1/8}{8 - 1/8} = \frac{65/8}{63/8} = \frac{65}{63}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the cosine of an acute angle, $\cos\theta = \frac{1}{8}$.
We need to calculate the value of the algebraic trigonometric expression:
\[ \frac{8\sec\theta + 1}{8\sec\theta - 1} \]

Step 2: Key Formula or Approach:
The secant function ($\sec\theta$) is the reciprocal of the cosine function ($\cos\theta$):
\[ \sec\theta = \frac{1}{\cos\theta} \]
We can compute the value of $\sec\theta$ directly and substitute it into the given expression.

Step 3: Detailed Explanation:

• Given:
\[ \cos\theta = \frac{1}{8} \]

• Find the value of $\sec\theta$ using the reciprocal identity:
\[ \sec\theta = \frac{1}{\cos\theta} = \frac{1}{\frac{1}{8}} = 8 \]

• Write down the target expression:
\[ \text{Expression} = \frac{8\sec\theta + 1}{8\sec\theta - 1} \]

• Substitute $\sec\theta = 8$ into the expression:
\[ \text{Expression} = \frac{8(8) + 1}{8(8) - 1} \]
\[ \text{Expression} = \frac{64 + 1}{64 - 1} \]
\[ \text{Expression} = \frac{65}{63} \]


Step 4: Final Answer:
The value of the expression is $\frac{65}{63}$.