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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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### Solving Inequalities - a curious observation

I'm a little unclear about your question, as it mentions two critical values (I'm assuming you mean restrictions or non-permissible values) but the inequality you included only has one. However, here ...
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