<s>[INST] Unsure if I have solved/proven this trigonometric inequality. This is my first post here. I apologize if it goes against any guidelines for posting. I study math as a hobby and am currently dealing with trigonometry on a high school level. I have so far learned the formulas for trigonometric addition and subtraction and double the angle, as well as what is in my language referred to as the ’trigonometric one’ - getting the radius of the unit circle by use of the pythagorean theorem. I have not yet gotten to deriving trigonometric functions. The following is a problem I could solve by plugging in a set of numbers, but in seeking a more elegant solution, perhaps, I found myself stuck and I don’t know what I am missing. I am appreciative of any help I get. The problem is as follows:

Show that if $A$ is an angle and $0^\circ<A<90^\circ$ then $\hspace{0.3cm}\left( 1+\dfrac {1}{\sin A}\right) \left( 1+\dfrac {1}{\cos A}\right)>5$

I began with the following assumption:
$$0^\circ<A<90^\circ\rightarrow0<{\sin A}<1\\0<{\cos A}<1\rightarrow\dfrac {1}{\sin A}\\\dfrac {1}{\cos A}>1$$
Given the above, it would follow that:
$$\begin{aligned}
\lim _{A\rightarrow 90^\circ}\dfrac {1}{\cos A}&=\infty \\
\lim _{A\rightarrow 0^\circ}\dfrac {1}{\sin A}&=\infty
\end{aligned}$$
This alone doesn’t seem like enough to show what is asked. I can show that at $A=45^\circ$ the product is still greater than 5, but I am not sure how any offset in degrees from there affects two trigonometric terms such that the product is still greater than 5. I also tried solving the inequality but ended up with fractioned terms I couldn’t add up or a cubic function if you will, that I couldn’t solve.
 [/INST] Use $AM-GM$ and $\sin^2 A + \cos^2 A = 1$.
.....
I start by simply doing.
$(1 + \frac {1}{\sin A})(1 + \frac 1{\cos A}) = $
$(1 + \frac 1{\sin A} + \frac 1{\cos A} + \frac 1{\sin A \cos A}$.
Now $0 < \sin A < 1$ so $ \frac 1{\sin A} > 1$ and $0 < \cos A < 1$ so $ \frac 1{\cos A} > 1$.
So $(1 + \frac 1{\sin A} + \frac 1{\cos A} + \frac 1{\sin A \cos A} > 3 + + \frac 1{\sin A \cos B}$.
Now common sense tells us $\sin A \cos A < 1*1$ so $\frac 1{\sin A \cos A} > 1$ and that gives us:
$(1 + \frac 1{\sin A} + \frac 1{\cos A} + \frac 1{\sin A \cos A}) > 4$ and that is not good enough.
So I need more.  Now I know $\sin A$ and $\cos A$ aren't just any numbers less than one.  I know $\sin^2 A + \cos^2 A = 1$ and I know that if gets close to $1$ or $0$ the other gets close to $0$ and $1$ and I know that $\sin 45 = \cos 45 = \frac{\sqrt {2}}{2}$ but other wise one is more and the other is less than $\frac{\sqrt{2}}{2}$.
This all triggers that I can probably use the AM-GM theorem to "boost" $\frac 1 {\sin A\cos b} > 1$ to $\frac 1{\sin A\cos B} > 2$.  Maybe... I'll use AM-GM and see what happens.
So we need to use AM-GM to prove $\sin A \cos B < \frac 12$ and $\frac 1{\sin A \cos A}> 2$.
Now AM- GM states for $M,N > 0$ thatn $\frac {M+N}2 \ge \sqrt{MN}$ so $\sin A\cos A = \sqrt{\sin^2 A\cos^2 A} \le \frac {\sin^2 A + \cos^2 A}2 = \frac 12$.
ANd that's that.
===
Another way $\cos A = \sin (90 - A)$.  Let $A = 45 + b$ then
$\sin A = \sin (45+b) = \sin 45 \cos b + \cos 45 \sin b=\frac {\sqrt 2}2(\cos b + \sin b)$ and $\cos A = \sin (45 - b) = \sin 45 \cos b - \cos 45 \sin b=\frac {\sqrt 2}2(\cos b - \sin b)$.
And $\sin A*\cos A = (\frac {\sqrt 2}2)^2(\cos b + \sin b)(\cos b - \sin b) = \frac 12(\cos^2 b - \sin^2 b)= \frac 12(1-2\sin^2 b) < \frac 12$
So $(1 +\frac 1{\sin A})(1 + \frac 1 {\cos A}) =$
$1 + \frac 1{\sin A} + \frac 1{\cos A} + \frac 1 {\sin A\cos B} > 1 + 1+1+2 =5$.
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