Animated Solution for Mathematics - Trigonometry: Prove that $\tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\cot 8\alpha =\cot \alpha$.

Unveiling the Telescoping Series

Welcome, future engineers. Today, we are not just solving a trigonometric identity; we are witnessing a mathematical ballet.

When you look at the expression $\tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\cot 8\alpha =\cot \alpha$, do not let the complexity intimidate you. Instead, look for the rhythm.

Notice how the angles double: $\alpha ,2\alpha ,4\alpha ,8\alpha$. Notice how the coefficients double: $1,2,4,8$. This is the signature of a telescoping series—a problem designed to collapse in on itself like a folding telescope.

The Master Key

To unlock this, we need a bridge. We need a way to relate $\tan \theta$ to $\cot 2\theta$. Let us derive this identity from first principles.

Consider the difference $\cot \theta -\tan \theta$. If we express these in terms of sine and cosine, we get:

Taking the common denominator, we arrive at:

Here is where the beauty of trigonometry shines. The numerator is the double-angle identity for cosine, $\cos 2\theta$. The denominator is half of the double-angle identity for sine, $\frac{1}{2}\sin 2\theta$.

Thus, our expression simplifies to:

We have found our master key: $\cot \theta -\tan \theta =2\cot 2\theta$. Rearranging this gives us the tool we need to dismantle the problem: $\tan \theta =\cot \theta -2\cot 2\theta$.

The Telescoping Dance

Now, let us apply this key to our original expression. We start with the first term, $\tan \alpha$. Using our identity, we replace it:

Look at the first two terms: $-2\cot 2\alpha +2\tan 2\alpha$. If we factor out a $-2$, we get $-2(\cot 2\alpha -\tan 2\alpha )$.

Does that bracket look familiar? It is our identity again, but with $2\alpha$ instead of $\alpha$! Since $\cot 2\alpha -\tan 2\alpha =2\cot 4\alpha$, our expression becomes:

The Final Cancellation

We are almost there. The pattern is repeating. We now have $-4\cot 4\alpha +4\tan 4\alpha$. Again, factor out the $-4$:

Applying our identity one last time, where $\cot 4\alpha -\tan 4\alpha =2\cot 8\alpha$, the expression transforms into:

And there it is. The $-8\cot 8\alpha$ and the $+8\cot 8\alpha$ vanish into thin air, leaving us with exactly $\cot \alpha$. We have arrived at the Right Hand Side.

This is the elegance of mathematics—taking a complex, intimidating string of terms and watching them simplify into a single, clean result. Keep this telescoping technique in your toolkit; it is a powerful weapon for your JEE Advanced journey.