Use the identity \(\cot\theta - \tan\theta = 2\cot 2\theta\), that is, \(\tan\theta = \cot\theta - 2\cot 2\theta\).

• \(\tan\theta = \cot\theta - 2\cot 2\theta\)
• \(2\tan 2\theta = 2\cot 2\theta - 4\cot 4\theta\)
• \(4\tan 4\theta = 4\cot 4\theta - 8\cot 8\theta\)

Adding these three, the middle terms cancel: \(\tan\theta + 2\tan 2\theta + 4\tan 4\theta = \cot\theta - 8\cot 8\theta\).

Add \(8\cot 8\theta\) to both sides. The total is \(\cot\theta\).

Answer: (A) \(\cot\theta\)