frac{{sqrt{2} - sin alpha - cos alpha }}{{sin | vedclass

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Question

(C) Given expression: $E = \frac{{\sqrt{2} - (\sin \alpha + \cos \alpha )}}{{\sin \alpha - \cos \alpha }}$Multiply numerator and denominator by $\frac

Vedclass · Accepted Answer

$	an \left( {\frac{\alpha }{2} - \frac{\pi }{8}} ight)$

Vedclass · Answer

(C) Given expression: $E = \frac{{\sqrt{2} - (\sin \alpha + \cos \alpha )}}{{\sin \alpha - \cos \alpha }}$Multiply numerator and denominator by $\frac{1}{\sqrt{2}}$:$E = \frac{{1 - (\frac{1}{\sqrt{2}}\sin \alpha + \frac{1}{\sqrt{2}}\cos \alpha )}}{{\frac{1}{\sqrt{2}}\sin \alpha - \frac{1}{\sqrt{2}}\cos \alpha }}$Using $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$:$E = \frac{{1 - \cos(\alpha - \frac{\pi}{4})}}{{\sin(\alpha - \frac{\pi}{4})}}$Let $	heta = \alpha - \frac{\pi}{4}$. Then $E = \frac{{1 - \cos 	heta }}{{\sin 	heta }}$Using half-angle formulas $1 - \cos 	heta = 2\sin^2(\frac{	heta}{2})$ and $\sin 	heta = 2\sin(\frac{	heta}{2})\cos(\frac{	heta}{2})$:$E = \frac{{2\sin^2(\frac{	heta}{2})}}{{2\sin(\frac{	heta}{2})\cos(\frac{	heta}{2})}} = 	an(\frac{	heta}{2})$Substituting $	heta = \alpha - \frac{\pi}{4}$ back:$E = 	an(\frac{\alpha}{2} - \frac{\pi}{8})$

$\frac{{\sqrt{2} - \sin \alpha - \cos \alpha }}{{\sin \alpha - \cos \alpha }} = $

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