For 0 < \theta < \frac{\pi}{2}, consider the following : I. (\tan^4 \theta + \tan^6 \theta)(\cot^4 \theta + \cot^6 \theta) = \sec^2 \theta \operatorname{cosec}^2 \theta II. \frac{\tan \theta + \sin \theta}{\tan \theta - \sin \theta} = \cot^2 \theta (\sec \theta + 1)^2 Which of the above is/are identities ?

  1. A. I only
  2. B. II only
  3. C. Both I and II ✓
  4. D. Neither I nor II

Correct Answer: C. Both I and II

Explanation

For I: \tan^4\theta(1+\tan^2\theta)\cot^4\theta(1+\cot^2\theta) = \tan^4\theta\sec^2\theta\cot^4\theta\operatorname{cosec}^2\theta = \sec^2\theta\operatorname{cosec}^2\theta. For II: LHS simplifies to \frac{\sec\theta+1}{\sec\theta-1}, and RHS simplifies to \frac{1}{\sec^2\theta-1}(\sec\theta+1)^2 = \frac{(\sec\theta+1)^2}{(\sec\theta-1)(\sec\theta+1)} = \frac{\sec\theta+1}{\sec\theta-1}. Both are true.

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