Question

If \(\theta=\frac{11\pi}{7}\), then \[ \frac{1+\cos 8\theta}{\cot^{2}4\theta} + \frac{1-\cos 8\theta}{\tan^{2}4\theta} = \]

• A. \(\sin\frac{\pi}{7}\)
• B. \(\cos\frac{2\pi}{7}\)
• C. \(2\)
• D. \(0\)

Hint:

Convert all expressions into \(\sin\) and \(\cos\) before substituting numerical values of angles.

Answer 1

The Correct Option is C

Concept: Use \[ 1+\cos2A=2\cos^2A \] and \[ 1-\cos2A=2\sin^2A. \]

Step 1: Apply identities.
\[ 1+\cos8\theta = 2\cos^24\theta \] \[ 1-\cos8\theta = 2\sin^24\theta. \] Substituting, \[ \frac{2\cos^24\theta}{\cot^24\theta} + \frac{2\sin^24\theta}{\tan^24\theta}. \]

Step 2: Use definitions of \(\tan\) and \(\cot\).
\[ = 2\sin^24\theta + 2\cos^24\theta. \] \[ = 2(\sin^24\theta+\cos^24\theta). \] \[ =2. \] \[ \boxed{2} \]