cos-x-sin-x-1-sin-2x-sec-2x-tan-2x-prove-it-

Question Number 87069 by jagoll last updated on 02/Apr/20

\frac{\cos x - \sin x}{\sqrt{1 + \sin 2 x}} = \sec 2 x - \tan 2 x

prove it

Commented by jagoll last updated on 02/Apr/20

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

= \sin x + \cos x

\Rightarrow \frac{\cos x - \sin x}{\cos x + \sin x} = \frac{{(\cos x - \sin x)}^{2}}{\cos^{2} x - \sin^{2} x}

= \frac{1 - \sin 2 x}{\cos 2 x} = \frac{1}{\cos 2 x} - \frac{\sin 2 x}{\cos 2 x}

= \sec 2 x - \tan 2 x

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