<s>[INST] Determine the value of the expression $\frac {5}{\sin^2A} + \frac{7}{\cos^2B}$ if $\cos A = \frac{-2}{3}$ and $\sin B=\frac{3}{4}$ 
Determine the value of the expression $\frac {5}{\sin^2A} + \frac{7}{\cos^2B}$ if $\cos A = \frac{-2}{3}$ and $\sin B=\frac{3}{4}$.

I applied $$\cos^2B\;=\;1-\sin^2B,$$
and $$\tan^2A\;=\;\frac{\sin^2A}{\cos^2A}.$$ But I am not sure what to do from there.
 [/INST] Using this trigononometric identity
$$ \cos^{2}\theta+\sin^{2}\theta =1 $$
we get that
$$ \cos^{2}B = 1-\sin^{2}B $$
similary we can find $\sin^{2}A$, and then we just replace
$$   \frac{5}{1-\cos^{2}A} + \frac{7}{1-\sin^{2}B}=\frac{5}{1-\left(\frac {-2}{3}\right)^{2}} + \frac{7}{1-\left(\frac {3}{4}\right)^{2}} $$
there you go...
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