Question

If $\frac{1-\tan x}{1+\tan x}=\tan y$ and $x-y=\frac{\pi }{6}$ , then $x,y$ are

• A. $\frac{5\pi }{24},\frac{\pi }{24}$
• B. $-\frac{7\pi }{24},-\frac{11\pi }{24}$
• C. $-\frac{115\pi }{24},-\frac{119\pi }{24}$
• D. All the above

Answer 1

The correct option is D All the above

From the second equation, $y=x-\frac{\pi }{6}$
From first equation,
$\tan y=\frac{1-\tan x}{1+\tan x}$
Or,$y=n\pi +\frac{\pi }{4}-x$
So, $x+y=n\pi +\frac{\pi }{4}$
Thus, the obtained values of $x$ and $y$ should satisfy both the equations ie
$x-y=\frac{\pi }{6}$ and $x+y=n\pi +\frac{\pi }{4}$
Since, all the three options satisfy these equations.
Hence, option 'D' is correct.