Find period of $$\dfrac{{1 + \sin x}}{{\cos x\left( {1 + cosecx} \right)}}$$

Solution

$$\dfrac{1+\sin x}{\cos x(1+{\rm cosec}\,x)}$$

$$=\dfrac{1+\sin x}{\cos x\left(1+\dfrac{1}{\sin x}\right)}$$

$$=\dfrac{\sin x(1+\sin x)}{\cos x(1+\sin x)}$$

$$\tan x$$

$$\hbox{which has period }2\pi$$