How can I prove that this function is bounded using the given hint? $$f(x) = \frac{1-2\sin x}{1 + \cos^2 x}, $$
Hint:
- $|A + B| \leq |A| + |B|,$
- $|\sin x| \leq 1.$
2026-04-06 16:13:32.1775492012
How can I prove that this function is bounded using the given hint?
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$$\frac{1-2\sin{x}}{1+\cos^2x}\leq\frac{|1-2\sin{x}|}{1+\cos^2x}\leq\frac{|1|+|-2\sin{x}|}{1+\cos^2x}\leq\frac{3}{1+\cos^2x}\leq3.$$ $$\frac{1-2\sin{x}}{1+\cos^2x}\geq\frac{|1|-|2\sin{x}|}{1+\cos^2x}\geq\frac{1-2}{1}=-1.$$ Thus, $$-1\leq f(x)\leq3$$ and our function is indeed boubded.