Question

The value of \[ \cos\frac{5\pi}{17}+\cos\frac{7\pi}{17}+2\cos\frac{11\pi}{17}\cos\frac{\pi}{17} \] is

• A. \(0\)
• B. \(1\)
• C. \(-1\)
• D. \(\frac{1}{2}\)

Hint:

Use \(2\cos A\cos B=\cos(A+B)+\cos(A-B)\) and \(\cos(\pi-\theta)=-\cos\theta\).

Answer 1

The Correct Option is A

Step 1: Use identity: \[ 2\cos A\cos B=\cos(A+B)+\cos(A-B) \]

Step 2: Here, \[ 2\cos\frac{11\pi}{17}\cos\frac{\pi}{17} = \cos\frac{12\pi}{17}+\cos\frac{10\pi}{17} \]

Step 3: So expression becomes: \[ \cos\frac{5\pi}{17}+\cos\frac{7\pi}{17} +\cos\frac{12\pi}{17}+\cos\frac{10\pi}{17} \]

Step 4: Use: \[ \cos(\pi-\alpha)=-\cos\alpha \] \[ \cos\frac{12\pi}{17}=\cos\left(\pi-\frac{5\pi}{17}\right)=-\cos\frac{5\pi}{17} \] \[ \cos\frac{10\pi}{17}=\cos\left(\pi-\frac{7\pi}{17}\right)=-\cos\frac{7\pi}{17} \]

Step 5: Hence all terms cancel: \[ 0 \] \[ \boxed{0} \]