Question

What is the general solution of the equation \( \cot\theta + \tan\theta = 2 \)? 

• A. \( \theta = n\pi/2 + \pi/4 \)
• B. \( \theta = n\pi/2 + \pi/6 \)
• C. \( \theta = n\pi + \pi/4 \)
• D. \( \theta = n\pi/2 + \pi/8 \)

Answer 1

The Correct Option is C

Step 1: Write \( \cot\theta + \tan\theta \) in terms of \(\sin\theta\) and \(\cos\theta\). \[ \cot\theta + \tan\theta = \frac{\cos\theta}{\sin\theta} + \frac{\sin\theta}{\cos\theta} = \frac{\cos^2\theta + \sin^2\theta}{\sin\theta \cos\theta} = \frac{1}{\sin\theta \cos\theta} \] Given that: \[ \frac{1}{\sin\theta \cos\theta} = 2 \] Step 2: Simplify. \[ \sin\theta \cos\theta = \frac{1}{2} \] We know that \( \sin 2\theta = 2 \sin\theta \cos\theta \), so: \[ \sin 2\theta = 2 \times \frac{1}{2} = 1 \] Step 3: Solve for \(2\theta\): \[ 2\theta = \frac{\pi}{2} + 2n\pi \] Therefore, \[ \theta = \frac{\pi}{4} + n\pi \] Final Answer: \[ \boxed{\theta = n\pi + \frac{\pi}{4}} \] Correct Option: (c)