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Without any software and approximations prove that $\sec(52^{\circ})-\cos(52^{\circ})>1$

We have that for $\theta\in(0,\pi/2)$, $f(\theta)=\sec \theta -\cos \theta$ is an increasing function and $$\sec \theta-\cos \theta =1 \implies \cos \theta = \frac{\sqrt 5-1}2 =\frac 1\varphi$$ that ...
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A question on a property of definite integrals that $\int_a^b f(x)dx =\int_a^b f(a+b-x)dx$

Let $$I=\int_\frac{\pi}{6}^\frac{\pi}{3}\frac{1}{1+\sqrt{(\tan(x)}}dx$$ Using the substitution $t=({\frac{\pi}{3} + \frac{\pi}{6}})-x=\frac{\pi}{2}-x\implies dt=-dx$ I=\int_\frac{\pi}{6}^\frac{\pi}{...
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Stable set for $f(x) = \frac{\pi}{2}\sin(x)$

Assuming your angles are in $(-\pi, \pi]$, there are two stable points: $\pm \pi/2$ (and one unstable point: $0$). This can be seen by plotting $y=x$ and $y=f(x)=\frac{\pi}{2}\sin x$ on the same graph....
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