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

$$\frac{\cos \mathrm{x}-\sin \mathrm{x}}{\sqrt{1+\sin 2\mathrm{x}}}=\sec 2\mathrm{x}-\tan 2\mathrm{x}$$$$\mathrm{prove}\mathrm{it}$$

Commented by jagoll last updated on 02/Apr/20

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

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