Find number of solution of : $$\sin x+\cos x =k\tan x \ \ \ \ :k \in \mathbb{R} , 0\leq x\leq2\pi$$
My work :
$$\sqrt{2}\sin (x+\pi/4)=\frac{k}{\sqrt{2}}\tan x$$
now $\dfrac{k}{\sqrt{2}}=\tan y$ then :
$$\sin (x+\pi/4)=\tan y \tan x$$
now what do i do ?
HINT
Consider the function
$$f(x)=\frac{\sin x+\cos x}{\tan x}=\frac{\sqrt{2}\sin (\frac{\pi}{4}+x)}{\tan x}$$
and study the number of solution of
$$f(x)=k$$