Question

If $\sec x=\sqrt{2}$ and $\frac{3\pi }{2}<x<2\pi$, find the value of $\frac{1+\tan x+cosecx}{1+\cot x-cosecx}$

Answer 1

Given that,

$\sec x=\sqrt{2}$ and $\frac{3\pi }{2}<x<2\pi$

$\sec x=\sqrt{2}=\sec {315}^{\circ }$

$\therefore x={315}^{\circ }$

Now, $\tan x=\tan {315}^{\circ }=-1$

$\cot x=\cot {315}^{\circ }=-1$

$cosecx=cosec{315}^{\circ }=-\sqrt{2}$

$\frac{1+\tan x+cosecx}{1+\cot x-cosecx}=\frac{1-1-\sqrt{2}}{1-1-(-\sqrt{2})}=\frac{-\sqrt{2}}{\sqrt{2}}=-1$