• question_answer The value of $\frac{\tan x}{\tan \,3x}$whenever defined never lie between [Kurukshetra CEE 1998; IIT 1992] A) 1/3 and 3 B) 1/4  and 4 C) 1/5 and 5 D) 5 and 6

Let $y=\frac{\tan x}{\tan 3x}=\frac{\tan x}{\frac{3\tan x-{{\tan }^{3}}x}{1-3{{\tan }^{2}}x}}$ $y=\frac{1-3{{\tan }^{2}}x}{3-{{\tan }^{2}}x}=\frac{\frac{1}{3}-{{\tan }^{2}}x}{1-\frac{1}{3}.{{\tan }^{2}}x}$ Hence, y should never lie between $\frac{1}{3}$and 3 whenever defined.