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I know that the True Negative Rate plus the False Positive Rate is equal to 1. However, I don't quite see how the True Positive Rate and False Positive Rate are related. Is there a relation between them?

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It might be helpful to think of these quantities as conditional probabilities. For concreteness, consider a medical test used to determine whether a patient has a disease. Define the following events: $$\begin{align*} "+" &= \text{the test is positive} \\ "-" & = \text{the test is negative} \\ \text{Disease} & = \text{the patient has the disease} \\ \text{Healthy} & = \text{the patient does not have the disease} \\ \end{align*}$$

Then $$\begin{align*} \text{True Negative Rate} &= P(- \mid \text{Healthy}) \\ \text{False Positive Rate} &= P(+ \mid \text{Healthy}) \end{align*}$$

Since the events $"+"$ and $"-"$ are the only possible outcomes given a healthy patient, these probabilities must sum to 1, as you observe.

On the other hand

$$\begin{align*} \text{True Positive Rate} &= P(+ \mid \text{Disease}) \end{align*}$$

conditions on the event $\text{Disease}$, which is disjoint from $\text{Healthy}$. So in general there is nothing we can say about the relationship between the True and False Positive Rates.

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As @tddevlin pointed out, there is no way to predict true positive rate from false positive rate, so they are not related.

However, if you know the actual numbers of true positives and false positives, you can quantify the precision of your classifier which would tell you the fraction of positive classifications that were actually true.

$$Precision = \frac{True Positives}{True Positives + False Positives}$$

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