Question

Prove that 1+sin2○+cos2○ /1+sin2○-cos2○ = cot​○

Answer 1

Dear Student,

Consider

$$\begin{gathered} \frac{1+\sin \left(2\theta \right)+\cos \left(2\theta \right)}{1+\sin \left(2\theta \right)-\cos \left(2\theta \right)} \\ =\frac{{\displaystyle [1+\cos \left(2\theta \right)]+\sin \left(2\theta \right)}}{{\displaystyle [1-\cos \left(2\theta \right)]+\sin \left(2\theta \right)}} \\ =\frac{2{\cos }^2\left(\theta \right)+2\sin \left(\theta \right)\cos \left(\theta \right)}{2{\sin }^2\left(\theta \right)+2\sin \left(\theta \right)\cos \left(\theta \right)} \\ =\frac{{\displaystyle 2\cos \left(\theta \right)(\cos \left(\theta \right)+\sin \left(\theta \right))}}{{\displaystyle 2\sin \left(\theta \right)(\sin \left(\theta \right)+\cos \left(\theta \right))}} \\ =\frac{\cos \left(\theta \right)}{\sin \left(\theta \right)} \\ =\cot \left(\theta \right) \end{gathered}$$

Regards