Question

$\frac{\sqrt{2}−\sin Î\pm −\cos Î\pm }{\sin Î\pm −\cos Î\pm }$ is equal to

• A. $\sec \left(\frac{Î\pm }{2}−\frac{Ï€}{8}\right)$
• B. $\cos \left(\frac{Ï€}{8}−\frac{Î\pm }{2}\right)$
• C. $\tan \left(\frac{Î\pm }{2}−\frac{Ï€}{8}\right)$
• D. $\cot \left(\frac{Î\pm }{2}−\frac{Ï€}{2}\right)$

Answer 1

Answer: (C)

$\frac{\sqrt{2}−\sin Î\pm −\cos Î\pm }{\sin Î\pm −\cos Î\pm }$
$=\frac{\sqrt{2}−\sqrt{2}\left\{\frac{1}{\sqrt{2}}\sin Î\pm +\frac{1}{\sqrt{2}}\cos Î\pm \right\}}{\sqrt{2}\left\{\frac{1}{\sqrt{2}}\sin Î\pm −\frac{1}{\sqrt{2}}\cos Î\pm \right\}}$
$=\frac{\sqrt{2}−\sqrt{2}\cos \left(Î\pm −\frac{Ï€}{4}\right)}{\sqrt{2}\sin \left(Î\pm −\frac{Ï€}{4}\right)}$
$=\frac{\sqrt{2}\left\{1−\cos θ\right\}}{\sqrt{2}\sin θ}$ , where $θ=Î\pm −\frac{Ï€}{4}$
$=\frac{2{\sin }^2\left(θ/2\right)}{2\sin \left(θ/2\right)\cos \left(θ/2\right)}=\tan \left(θ/2\right)$
$=\tan \left(\frac{Î\pm }{2}−\frac{Ï€}{8}\right)$