Question

The sum of all the values of \( \theta \in \left[\frac{\pi}{2}, 2\pi\right] \) satisfying the equation \( \sin^2\theta \tan\theta + \cos^2\theta \cot\theta = \sin 2\theta \) is:

• A. \( \frac{15\pi}{4} \)
• B. \( \frac{15\pi}{2} \)
• C. \( 7\pi \)
• D. \( \frac{11\pi}{2} \)

Hint:

The equation \( \sin^4\theta + \cos^4\theta = 2\sin^2\theta\cos^2\theta \) is a perfect square. Always look for algebraic identities like \( (a^2-b^2)^2 \) hidden within trigonometric powers to reduce the order of the equation.

Answer 1

The Correct Option is A

Concept: To solve trigonometric equations, simplify the terms into a single function (usually \( \tan \theta \)) and identify solutions within the specified interval.
• \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), \( \cot\theta = \frac{\cos\theta}{\sin\theta} \)
• \( \sin 2\theta = 2 \sin\theta \cos\theta \)

Step 1: Simplifying the given equation.
\[ \sin^2\theta \left(\frac{\sin\theta}{\cos\theta}\right) + \cos^2\theta \left(\frac{\cos\theta}{\sin\theta}\right) = 2\sin\theta\cos\theta \] \[ \frac{\sin^3\theta}{\cos\theta} + \frac{\cos^3\theta}{\sin\theta} = 2\sin\theta\cos\theta \] Multiply the entire equation by \( \sin\theta\cos\theta \): \[ \sin^4\theta + \cos^4\theta = 2\sin^2\theta\cos^2\theta \implies \sin^4\theta - 2\sin^2\theta\cos^2\theta + \cos^4\theta = 0 \] \[ (\sin^2\theta - \cos^2\theta)^2 = 0 \implies \sin^2\theta = \cos^2\theta \] Dividing by \( \cos^2\theta \): \( \tan^2\theta = 1 \implies \tan\theta = \pm 1 \).

Step 2: Finding solutions in the interval \( [\pi/2, 2\pi] \).
- \( \tan\theta = 1 \): \( \theta = \frac{5\pi}{4} \) (3rd quadrant). - \( \tan\theta = -1 \): \( \theta = \frac{3\pi}{4} \) (2nd quadrant) and \( \theta = \frac{7\pi}{4} \) (4th quadrant).

Step 3: Calculating the sum.
\[ \text{Sum} = \frac{3\pi}{4} + \frac{5\pi}{4} + \frac{7\pi}{4} = \frac{15\pi}{4} \]