Question

Trig Identities

Answer 1

Use the double angle formula\[\tan(2\Theta) = \frac{ 2\tan \theta }{ 1-\tan^2 \theta } \]

Answer 2

so \[\tan(2\theta) =1\]

Answer 3

Yup.

Answer 4

Alternatively, you can treat this as a quadratic:
\[\begin{align*}\frac{2\tan x}{1-\tan^2x}=1~~\Rightarrow~~&2\tan x=1-\tan^2x\\\\
&\tan^2x+2\tan x - 1=0\end{align*}\]
Substitute \(y=\tan x\), so you have
\[y^2+2y-1=0\\
y=\frac{-2\pm\sqrt{8}}{2}\\
y=-1\pm\sqrt2\\
\tan x=-1\pm\sqrt2\]
You'd need a calculator to determine the solution, so you're better off using the identity.