Question

For \(\theta\in\left(0,\frac{\pi}{2}\right)\), if the complete range of \[ (\cot^2\theta-\cos^2\theta)(\tan^2\theta-\sin^2\theta) \] is \((\alpha,\beta]\), then \(\beta-\alpha=\)

• A. \(1\)
• B. \(\dfrac{1}{2}\)
• C. \(\dfrac{1}{4}\)
• D. \(2\) \bigskip

Hint:

Whenever expressions contain both \(\tan\theta\) and \(\cot\theta\), rewrite them in terms of \(\sin\theta\) and \(\cos\theta\). Many complicated products reduce to powers of \(\sin\theta\cos\theta\), after which the identity \[ \sin2\theta=2\sin\theta\cos\theta \] makes finding the range very easy.

Answer 1

The Correct Option is C

Concept: For trigonometric range problems, first simplify the expression using standard identities and then convert it into a function of a single variable whose range can be obtained easily. Useful identities: \[ \cot^2\theta=\frac{\cos^2\theta}{\sin^2\theta}, \qquad \tan^2\theta=\frac{\sin^2\theta}{\cos^2\theta}. \]

Step 1: Simplify each factor. Let \[ E=(\cot^2\theta-\cos^2\theta)(\tan^2\theta-\sin^2\theta). \] Now, \[ \cot^2\theta-\cos^2\theta = \frac{\cos^2\theta}{\sin^2\theta}-\cos^2\theta. \] \[ = \cos^2\theta \left( \frac{1}{\sin^2\theta}-1 \right). \] \[ = \cos^2\theta \left( \frac{1-\sin^2\theta}{\sin^2\theta} \right). \] \[ = \frac{\cos^4\theta}{\sin^2\theta}. \] Similarly, \[ \tan^2\theta-\sin^2\theta = \frac{\sin^2\theta}{\cos^2\theta}-\sin^2\theta. \] \[ = \sin^2\theta \left( \frac{1}{\cos^2\theta}-1 \right). \] \[ = \sin^2\theta \left( \frac{1-\cos^2\theta}{\cos^2\theta} \right). \] \[ = \frac{\sin^4\theta}{\cos^2\theta}. \]

Step 2: Multiply the factors. Therefore, \[ E = \frac{\cos^4\theta}{\sin^2\theta} \cdot \frac{\sin^4\theta}{\cos^2\theta}. \] \[ = \sin^2\theta\cos^2\theta. \] Using \[ \sin2\theta=2\sin\theta\cos\theta, \] we get \[ E = \frac{1}{4}\sin^22\theta. \]

Step 3: Find the range of \(E\). Since \[ 0<\theta<\frac{\pi}{2}, \] we have \[ 0<2\theta<\pi. \] Hence, \[ 0<\sin^22\theta\le 1. \] Therefore, \[ 0<E\le \frac14. \] Thus the complete range is \[ (\alpha,\beta] = \left(0,\frac14\right]. \] So, \[ \alpha=0, \qquad \beta=\frac14. \]

Step 4: Compute \(\beta-\alpha\). \[ \beta-\alpha = \frac14-0. \] \[ = \frac14. \]

Step 5: Write the final answer. \[ \boxed{\frac14} \]