NIMCET 2013 — Mathematics PYQ

Solution

Step 1: Use the identity $\cot A-\tan A=2\cot 2A$

Rearranging this identity, we get:

$\tan A=\cot A-2\cot 2A$

Step 2: Substitute this identity into the expression

We will replace each $\tan$ term sequentially starting from the smallest angle.

• For $\tan \theta$:

  $$\tan \theta =\cot \theta -2\cot 2\theta$$

• Substitute this into the first part of the expression:

  $$(\cot \theta -2\cot 2\theta )+2\tan 2\theta$$
  $$\cot \theta -2(\cot 2\theta -\tan 2\theta )$$

• Using the identity $\cot 2\theta -\tan 2\theta =2\cot 4\theta$:

  $$\cot \theta -2(2\cot 4\theta )$$
  $$\cot \theta -4\cot 4\theta$$

Step 3: Continue with the next term

Now add the $4\tan 4\theta$ term:

• Using the identity $\cot 4\theta -\tan 4\theta =2\cot 8\theta$:

  $$\cot \theta -4(2\cot 8\theta )$$
  $$\cot \theta -8\cot 8\theta$$

Step 4: Add the final term

Finally, add the last term from the question ($8\cot 8\theta$):

Final Answer:

The value is $\cot \theta$, which corresponds to Option 1.