How should I have solved this?

Throughout I will use $\csc(x) = cosec(x)$ because I'm lazy

A good start will be to get a common denominator which will be $(\csc(x)+1)(\csc(x)-1)$ .

$\dfrac{\csc(x)(\csc(x)-1)}{(\csc(x)+1)} + \dfrac{\csc(x)(\csc(x)+1)}{\csc(x)-1} = \dfrac{\csc(x)(\csc(x)-1) + \csc(x)(\csc(x)+1)}{(\csc(x)-1)(\csc(x)+1)}$

When you expand you get (note that the denominator is difference of two squares)

$\dfrac{\csc^2(x)-1 + \csc^2(x)+1}{\csc^2(x)-1} = \dfrac{2\csc^2(x)}{\csc^2(x)-1}$

Spoiler:

Using the identities given we get $\dfrac{2\csc^2(x)}{\cot^2(x)} = \dfrac{2}{\sin^2(x)} \cdot \dfrac{\sin^2(x)}{\cos^2(x)}$

since we're dividing by a fraction I flipped it and multiplied. Cancelling gives

$\dfrac{2}{\cos^2(x)} = 2\sec^2(x)$

Set this equal to 50 and divide by 2 to finish the question

edit: don't want to give the whole working away so put some in a spoiler. Look if you want, tis no water off my back