Question

$\frac{\sin\frac{\pi}{7}+\sin\frac{2\pi}{7}}{1+\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}} = $ ________.

• A. $\cot\frac{\pi}{7}$
• B. $\cos\frac{\pi}{14}$
• C. $1+\sin\frac{\pi}{14}$
• D. $1+\cos\frac{\pi}{14}$
• E. $\tan\frac{\pi}{7}$

Hint:

$\frac{\sin A + \sin B}{\cos A + \cos B + 1}$ often relates to $\tan(\text{average angle})$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
The problem asks to simplify a complex trigonometric expression. The structure suggests using double angle and sum-to-product identities. A key observation is that one angle is double the other, i.e., \(\frac{2\pi}{7} = 2 \times \frac{\pi}{7}\).
Step 2: Key Formula or Approach
Let \(\theta = \frac{\pi}{7}\). The expression becomes \(\frac{\sin\theta + \sin(2\theta)}{1 + \cos\theta + \cos(2\theta)}\).
We will use the following double-angle identities:
1. \(\sin(2\theta) = 2\sin\theta\cos\theta\)
2. \(\cos(2\theta) = 2\cos^2\theta - 1\), which can be rearranged to \(1 + \cos(2\theta) = 2\cos^2\theta\).
Step 3: Detailed Explanation
1. Simplify the numerator.
The numerator is \(\sin\theta + \sin(2\theta)\).
Using the identity \(\sin(2\theta) = 2\sin\theta\cos\theta\), we get:
\[ \sin\theta + 2\sin\theta\cos\theta \] Factor out the common term \(\sin\theta\):
\[ \sin\theta(1 + 2\cos\theta) \] 2. Simplify the denominator.
The denominator is \(1 + \cos\theta + \cos(2\theta)\).
Let's rearrange the terms to group \(1\) and \(\cos(2\theta)\):
\[ (1 + \cos(2\theta)) + \cos\theta \] Using the identity \(1 + \cos(2\theta) = 2\cos^2\theta\), we get:
\[ 2\cos^2\theta + \cos\theta \] Factor out the common term \(\cos\theta\):
\[ \cos\theta(2\cos\theta + 1) \] 3. Combine the simplified numerator and denominator.
The original expression is now:
\[ \frac{\sin\theta(1 + 2\cos\theta)}{\cos\theta(1 + 2\cos\theta)} \] Assuming \(1 + 2\cos\theta \neq 0\) (which is true for \(\theta = \pi/7\)), we can cancel this common factor.
\[ \frac{\sin\theta}{\cos\theta} \] 4. Final Simplification.
We know that \(\frac{\sin\theta}{\cos\theta} = \tan\theta\).
Substituting back \(\theta = \frac{\pi}{7}\), the expression simplifies to:
\[ \tan\frac{\pi}{7} \] Step 4: Final Answer
The value of the expression is \(\tan\frac{\pi}{7}\).