Find the general solution of the equation $ an(x) = cot(2x) $ on the unit circle

To solve the equation $ \tan(x) = \cot(2x) $ on the unit circle, we start by expressing $ \cot(2x) $ in terms of $ \tan(2x) $:

$ \cot(2x) = \frac{1}{\tan(2x)} $

The equation becomes:

$ \tan(x) = \frac{1}{\tan(2x)} $

Using the double-angle identity for tangent:

$ \tan(2x) = \frac{2 \tan(x)}{1 – \tan^2(x)} $

Substitute this back into the equation:

$ \tan(x) = \frac{1}{\frac{2 \tan(x)}{1 – \tan^2(x)}} $

Simplify the equation:

$ \tan(x) = \frac{1 – \tan^2(x)}{2 \tan(x)} $

Rearrange the equation:

$ 2 \tan^2(x) = 1 – \tan^2(x) $

Combine like terms:

$ 3 \tan^2(x) = 1 $

Solve for $ \tan(x) $:

$ \tan(x) = \pm \frac{1}{\sqrt{3}} $

Therefore, the general solution on the unit circle is:

$ x = n\pi + (-1)^n \frac{\pi}{6} $ where $ n $ is an integer.

To solve the equation $ an(x) = cot(2x) $, rewrite $ cot(2x) $:

$ cot(2x) = frac{1}{ an(2x)} $

Then:

$ an(x) = frac{1}{frac{2 an(x)}{1 – an^2(x)}} $

Simplify:

$ an(x) = frac{1 – an^2(x)}{2 an(x)} $

Which leads to:

$ 2 an^2(x) = 1 – an^2(x) $

Combining terms:

$ 3 an^2(x) = 1 $

Solving for $ an(x) $:

$ an(x) = pm frac{1}{sqrt{3}} $

Thus, the solution is:

$ x = npi + (-1)^n frac{pi}{6}, n in mathbb{Z} $

Rewriting the equation $ an(x) = cot(2x) $ gives:

$ an(x) = frac{1}{ an(2x)} $

Using the identity:

$ an(2x) = frac{2 an(x)}{1 – an^2(x)} $

We obtain:

$ an(x) = frac{1 – an^2(x)}{2 an(x)} $

Which simplifies to:

$ an(x) = pm frac{1}{sqrt{3}} $

So the solution is:

$ x = npi + (-1)^n frac{pi}{6} $

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