How do you prove the identity tan2XcotX=3?

Oct 7, 2015

It isn't.

Explanation:

$\tan \left(2 x\right) \cdot \cot \left(x\right) = 3$

Using the double angle formula,

$\frac{2 \tan \left(x\right)}{1 - {\tan}^{2} \left(x\right)} \cdot \cot \left(x\right) = 3$

Knowing that $\tan \left(x\right) \cdot \cot \left(x\right) = 1$

$\frac{2}{1 - {\tan}^{2} \left(x\right)} = 3$

Which is obviously false, as the tangent range from $- \infty$ to $\infty$.
If you continue to work this like it was an equation, you'll see this only has two tangent values for solutions, $\pm \frac{\sqrt{3}}{3}$.