# Simplify (1+cos2A+sin2A)/(1-cos2A+sin2A)?

Sep 18, 2016

#### Explanation:

$\frac{1 + \cos 2 A + \sin 2 A}{1 - \cos 2 A + \sin 2 A}$

Now $\sin 2 A = 2 \sin A \cos A$ and $\cos 2 A = 2 {\cos}^{2} A - 1 = 1 - 2 {\sin}^{2} A$

Hence $\frac{1 + \cos 2 A + \sin 2 A}{1 - \cos 2 A + \sin 2 A}$

= $\frac{1 + 2 {\cos}^{2} A - 1 + 2 \sin A \cos A}{1 - \left(1 - 2 {\sin}^{2} A\right) + 2 \sin A \cos A}$

= $\frac{1 + 2 {\cos}^{2} A - 1 + 2 \sin A \cos A}{1 - 1 + 2 {\sin}^{2} A + 2 \sin A \cos A}$

= $\frac{2 {\cos}^{2} A + 2 \sin A \cos A}{2 {\sin}^{2} A + 2 \sin A \cos A}$

= $\frac{2 \cos A \left(\cos A + \sin A\right)}{2 \sin A \left(\sin A + \cos A\right)}$

= $\frac{2 \cos A}{2 \sin A}$

= $\cot A$