ساده کردن عبارت $ \frac{\cos 2x}{ \sqrt{2}\sin x -1} - \frac{{ \sqrt{2} \cos 2x} }{\sin x+\cos x} + \sqrt{2}\cos x$

Question

عبارت زير را ساده كنيد.!!

$$A:= \frac{\cos 2x}{ \sqrt{2}\sin x -1} - \frac{{ \sqrt{2} \cos 2x} }{\sin x+\cos x} + \sqrt{2}\cos x $$

Answer 1

$$A:= \frac{\cos 2x}{ \sqrt{2}\sin x -1} - \frac{{ \sqrt{2} \cos 2x} }{\sin x+\cos x} + \sqrt{2}\cos x $$

$$= \frac{1-2 \sin^{2} x}{ \sqrt{2}\sin x -1} - \frac{ \sqrt{2}( \cos^{2}x - \sin^{2}x ) }{\sin x+\cos x}+ \sqrt{2} \cos x $$

$$ =\frac{(1- \sqrt{2}\sin x )(1+ \sqrt{2}\sin x )}{-(1- \sqrt{2}\sin x )} - \frac{ \sqrt{2}(\cos x-\sin x)(\cos x+\sin x) }{(\cos x+\sin x)} + \sqrt{2} \cos x$$

$$=-(1+ \sqrt{2} \sin x)- \sqrt{2} (\cos x-\sin x)+ \sqrt{2} \cos x$$

$$=-1- \sqrt{2} \sin x- \sqrt{2} \cos x+\sqrt{2} \sin x + \sqrt{2} \cos x=-1$$

از اين روابط براي ساده كردن عبارت استفاده شده است :

$$ \cos 2x=1-2 \sin^{2} x= \cos^{2}x - \sin^{2} x$$

اتحاد مزدوج :

$$(a+b)(a-b)=( a^{2} - b^{2} )$$