Solve tan 2θ = tan (2/θ).

Step-by-step solution

1. 1

   Given equation

   Humein equation di gayi hai: $\tan 2\theta = \tan \left(\frac{2}{\theta}\right)$.

   $$\tan 2\theta = \tan \left(\frac{2}{\theta}\right)$$
2. 2

   General solution of tan

   General solution for $\tan x = \tan y$ is $x = n\pi + y$, where $n \in \mathbb{Z}$.

   $$x = n\pi + y$$
3. 3

   Apply general solution

   Is principle ko apply karte hain given equation par.

   $$2\theta = n\pi + \frac{2}{\theta}$$
4. 4

   Rearrange into quadratic

   Equation ko rearrange karke quadratic form mein likhte hain.

   $$2\theta^2 - n\pi\theta - 2 = 0$$
5. 5

   Solve quadratic for $\theta$

   Quadratic formula use karke $\theta$ ke liye solve karte hain.

   $$\theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
6. 6

   Substitute and simplify

   Values substitute karke simplify karte hain.

   $$\theta = \frac{n\pi \pm \sqrt{(n\pi)^2 - 4(2)(-2)}}{2(2)} = \frac{n\pi \pm \sqrt{n^2\pi^2 + 16}}{4}$$
7. 7

   Final solutions for $\theta$

   The solutions are $\theta = \frac{n\pi \pm \sqrt{n^2\pi^2 + 16}}{4}$ for $n \in \mathbb{Z}$.

   $$\theta = \frac{n\pi \pm \sqrt{n^2\pi^2 + 16}}{4}$$