Without any software and approximations prove that $\sec(52^{\circ})-\cos(52^{\circ})>1$ Without any software and approximations prove that $$\sec(52^{\circ})-\cos(52^{\circ})>1$$
We can use some known trig values like $18^{\circ}$,$54^{\circ}$,etc
My try:
I considered the function:
$$f(x)=\sec(x)-\cos(x)-1,\: x\in \left (0, \frac{\pi}{3}\right)$$
We have the derivative as:
$$f'(x)=\sec x\tan x+\sin x >0$$ so $f$ is Monotone increasing.
So we have:
$$f(52^{\circ})>f(45^{\circ})=\frac{1}{\sqrt{2}}-1$$ but not able to proceed

Hint .
Can you show the inequality for $10\leq x\leq 65$:
$$f\left(x\right)=x\left(\sec\left(x\cdot\frac{\pi}{180}\right)-\cos\left(\frac{\pi}{180}\cdot x\right)\right)> \frac{2(x-9)^{3}-2(x-9)^{2}+15(x-9)+75}{3000}$$
?
Some other hint :
Define :
$$h(x)=\left(\sec\left(x\cdot\frac{\pi}{180}\right)-\cos\left(\frac{\pi}{180}\cdot x\right)\right),p(x)=\frac{2(x-9)^{3}-2(x-9)^{2}+15(x-9)+75}{3000}$$
Then we have for $10<x<65$ :
$$h''(x)>0,p''(x)>0$$
So we have using strong convexity for $x\in[45,52]$ :
$$xh(x)\geq x\left(h'\left(45\right)\left(x-45\right)+h\left(45\right)+\frac{h''\left(45\right)}{2}\left(x-45\right)^{2}\right)>p(x)$$



If $0<a<1$ and $0\leq x\leq 2a\pi$ then :
$$1-\cos\left(x\right)-\left(\frac{\sin\left(a\pi\right)}{a\pi}\right)^{2}\cdot\frac{x^{2}}{2}\geq 0$$
See [1] for a reference .
Now a lemma using the concavity of $\sin(x)$ on $(0,\pi/2)$ :
We have :
$$0<\frac{\sin\left(\frac{\pi}{6.25}\right)-\sin\left(\frac{\pi}{6}\right)}{\frac{\pi}{6.25}-\frac{\pi}{6}}-\frac{\sqrt{3}}{2}$$
A second lemma :
$$\pi<\frac{185}{100}\sqrt{3}$$
Using lemma 1 and 2 with $a=1/6.25$ and $x=52\cdot\frac{\pi}{180}$ for the first inequality we got :
$$\left(\frac{6.25}{\pi}\left(\frac{\sqrt{3}}{2}\left(\frac{1.85\sqrt{3}}{6.25}-\frac{1.85\sqrt{3}}{6}\right)+\frac{1}{2}\right)\right)^{2}\cdot\frac{\left(52\cdot\frac{\pi}{180}\right)^{2}}{2}=1934881/512000<1-\cos\left(52\cdot\frac{\pi}{180}\right)$$
Wich is sufficient to show the claim proposed in the comment @MartinR.
Reference :
[1] Becker,Michael and Lawrence E.Stark An extremal inequality for the Fourier coefficients of positive cosine polynomials ,Univ. Beograd .Publ. Elektrotehn Fak. Ser . Mat ., No.577-No.588 (1977)57-58.
