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Question Number 45844 by ahmadpat222@gmail.com last updated on 17/Oct/18

$$\mathrm{If}y=\frac{{\sec }^2\theta -\tan \theta }{{\sec }^2\theta +\tan \theta },\mathrm{then}$$

Commented by MJS last updated on 17/Oct/18

$$\mathrm{again},{\mathrm{what}}^{\prime }\mathrm{s}\mathrm{the}\mathrm{question}?$$$$y=\frac{{\sec }^2\theta -\tan \theta }{{\sec }^2\theta -l+\tan \theta }\Rightarrow y\in \mathbb{R}\mathrm{\forall }\theta \in \mathbb{R}$$$$y=\frac{{\sec }^2\theta -\tan \theta }{{\sec }^2\theta -l+\tan \theta }\Rightarrow y>0\mathrm{\forall }\theta \in \mathbb{R}$$$$y=0\iff \theta =\frac{z\pi }{4}\pm \frac{\mathrm{i}}{2}\ln (2+\sqrt{3})\wedge z\in \mathbb{Z}$$$$...\mathrm{zillions}\mathrm{of}\mathrm{possibilities}$$

Answered by MJS last updated on 17/Oct/18

$$y=\frac{1-\sin \theta \cos \theta }{1+\sin \theta \cos \theta }=\frac{2-\sin 2\theta }{2+\sin 2\theta }$$

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