trigonometry - Calculate $frac{1}{sin(x)} +frac{1}{cos(x)}$ if $sin(x)+cos(x)=frac{7}{5}$

If

$$\begin{equation} \sin(x) + \cos(x) = \frac{7}{5}, \end{equation}$$

then what's the value of

$$\begin{equation} \frac{1}{\sin(x)} + \frac{1}{\cos(x)}\text{?} \end{equation}$$

Meaning the value of $\sin(x)$, $\cos(x)$ (the denominator) without using the identities of trigonometry.

The function $\sin x+\cos x$ could be transformed using some trigonometric identities to a single function. In fact, WolframAlpha says it is equal to $\sqrt2\sin\left(x+\frac\pi4\right)$ and there also are some posts on this site about this equality. So probably in this way we could calculate $x$ from the first equation - and once we know $\sin x$ and $\cos x$, we can calculate $\frac1{\sin x}+\frac1{\cos x}$. Is there a simpler solution (perhaps avoiding explicitly finding $x$)?