Induction of inequality involving AP Prove by induction that
$$(a_{1}+a_{2}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)\geq n^{2}$$
where $n$ is a positive integer and $a_1, a_2,\dots, a_n$ are real positive numbers 
Hence, show that 
$$\csc^{2}\theta +\sec^{2}\theta +\cot^{2}\theta \geq 9\cos^{2}\theta$$
Please help me.
Thank you!

For the first part, we need a base case and an inductive step.
Base Case: suppose we have only one number, $a_1$. Then $$(a_1)\left(\frac1{a_1}\right)=1\leq 1$$
Inductive Step: By Isomorphism's work, this amounts to showing that
$$
a_n\left(\frac{1}{a_1} + \frac{1}{a_2} \cdots + \frac{1}{a_{n-1}}\right) + \frac1{a_n}(a_1 + a_2 \cdots + a_{n-1})\geq 2n-2
$$
Start by multiplying through to get
$$
\begin{align}
a_n\left(\frac{1}{a_1} + \frac{1}{a_2} \cdots + \frac{1}{a_{n-1}}\right) + \frac1{a_n}(a_1 + a_2 \cdots + a_{n-1}) & = \\
\left(\frac{a_n}{a_1} + \frac{a_n}{a_2} \cdots + \frac{a_n}{a_{n-1}}\right) + \left(\frac{a_1}{a_n} + \frac{a_2}{a_n} \cdots + \frac{a_{n-1}}{a_n}\right) &= \\
\left(\frac{a_n}{a_1} + \frac{a_1}{a_n}\right) +
\left(\frac{a_n}{a_2} + \frac{a_2}{a_n}\right) + \dots +
\left(\frac{a_n}{a_{n-1}} + \frac{a_{n-1}}{a_n}\right) &\geq \\
2+2+\dots+2
\end{align}
$$
