Verifying Trigonometric Identity I

We need to check if \((\tan^4\theta + \tan^2\theta)(\cot^4\theta + \cot^2\theta) = \sec^2\theta \operatorname{cosec}^2\theta\) holds true for \(0 < \theta < \frac{\pi}{2}\).

• Start with the Left-Hand Side (LHS):

  LHS \(= (\tan^4\theta + \tan^2\theta)(\cot^4\theta + \cot^2\theta)\)

• Factor out common terms:

  LHS \(= \tan^2\theta(\tan^2\theta + 1) \cdot \cot^2\theta(\cot^2\theta + 1)\)

• Apply trigonometric identities:

  Use \(1 + \tan^2\theta = \sec^2\theta\) and \(1 + \cot^2\theta = \operatorname{cosec}^2\theta\).

  LHS \(= \tan^2\theta (\sec^2\theta) \cot^2\theta (\operatorname{cosec}^2\theta)\)

• Rearrange and use \(\tan\theta\cot\theta = 1\):

  LHS \(= (\tan^2\theta \cot^2\theta) (\sec^2\theta \operatorname{cosec}^2\theta)\)

  Since \(\tan^2\theta \cot^2\theta = (\tan\theta \cot\theta)^2 = 1^2 = 1\),

  LHS \(= (1) (\sec^2\theta \operatorname{cosec}^2\theta) = \sec^2\theta \operatorname{cosec}^2\theta\)

• Conclusion for Identity I:

  The LHS equals the Right-Hand Side (RHS). Thus, Identity I is valid.

Verifying Trigonometric Identity II

We need to check if \(\frac{\tan\theta+\sin\theta}{\tan\theta-\sin\theta} = \cot^2\theta(\sec\theta+1)^2\) holds true for \(0 < \theta < \frac{\pi}{2}\).

• Simplify the Left-Hand Side (LHS):

  LHS \(= \frac{\tan\theta+\sin\theta}{\tan\theta-\sin\theta}\)

  Substitute \(\tan\theta = \frac{\sin\theta}{\cos\theta}\): LHS \(= \frac{\frac{\sin\theta}{\cos\theta}+\sin\theta}{\frac{\sin\theta}{\cos\theta}-\sin\theta}\)

  Factor out \(\sin\theta\) (note \(\sin\theta \neq 0\) since \(0 < \theta < \frac{\pi}{2}\)): LHS \(= \frac{\sin\theta(\frac{1}{\cos\theta}+1)}{\sin\theta(\frac{1}{\cos\theta}-1)} = \frac{\frac{1}{\cos\theta}+1}{\frac{1}{\cos\theta}-1}\)

  Substitute \(\sec\theta = \frac{1}{\cos\theta}\): LHS \(= \frac{\sec\theta+1}{\sec\theta-1}\)

  Multiply numerator and denominator by \((\sec\theta+1)\): LHS \(= \frac{(\sec\theta+1)(\sec\theta+1)}{(\sec\theta-1)(\sec\theta+1)} = \frac{(\sec\theta+1)^2}{\sec^2\theta-1}\)

  Use the identity \(\sec^2\theta - 1 = \tan^2\theta\): LHS \(= \frac{(\sec\theta+1)^2}{\tan^2\theta}\)

  Rewrite using \(\cot^2\theta = \frac{1}{\tan^2\theta}\): LHS \(= \cot^2\theta(\sec\theta+1)^2\)

• Conclusion for Identity II:

  The simplified LHS equals the Right-Hand Side (RHS). Thus, Identity II is valid.

Overall Conclusion

Both Identity I and Identity II are valid trigonometric identities for the given range of \(\theta\). Therefore, the correct option is that both are true.