For All Real Values of x, cot x − 2 cot 2x is Equal to What?
Question
For all real values of \(x\),
\[ \cot x – 2\cot 2x \]
is equal to
(a) \(\tan 2x\)
(b) \(\tan x\)
(c) \(-\cot 3x\)
(d) none of these
Solution
Use the identity:
\[ \cot 2x=\frac{\cot^2x-1}{2\cot x} \]
Therefore,
\[ 2\cot 2x = \frac{\cot^2x-1}{\cot x} \]
Now,
\[ \cot x-2\cot 2x = \cot x-\frac{\cot^2x-1}{\cot x} \]
\[ = \frac{\cot^2x-(\cot^2x-1)}{\cot x} \]
\[ = \frac{1}{\cot x} \]
\[ =\tan x \]
Final Answer
\[ \boxed{\tan x} \]
Hence, the correct option is (b) \(\tan x\).