To address the problem, let's analyze the provided equations and steps thoroughly:
### First Condition:
The first equation is:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
Simplifying, we have:
[tex]\[ \left(-\frac{8}{17}\right)^2 = \frac{64}{289} \][/tex]
So the first equation becomes:
[tex]\[ \frac{\frac{64}{289}}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
This simplifies to:
[tex]\[ \frac{64}{289 \cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
To verify, let's rewrite the right-hand side (RHS):
[tex]\[ \frac{64}{289 \cos^2 \theta} + 1 = \frac{64}{289 \cos^2 \theta} + \frac{289 \cos^2 \theta}{289 \cos^2 \theta} = \frac{64 + 225 \cos^2 \theta}{289 \cos^2 \theta} \][/tex]
For the equation to hold true, we should have:
[tex]\[ \frac{64 + 225 \cos^2 \theta}{289 \cos^2 \theta} = \frac{1}{\cos^2 \theta} \][/tex]
Thus, if:
[tex]\[ 64 + 225 \cos^2 \theta = 289 \cos^2 \theta \][/tex]
Then:
[tex]\[ 64 = 64 \][/tex]
This clearly holds true for all [tex]\(\theta\)[/tex].
### Second Condition:
The second equation to verify is:
[tex]\[ \left(-\frac{8}{17}\right)^2 + \cos^2 \theta = 1 \][/tex]
Simplifying the left-hand side (LHS):
[tex]\[ \left(-\frac{8}{17}\right)^2 = \frac{64}{289} \][/tex]
Thus, the equation becomes:
[tex]\[ \frac{64}{289} + \cos^2 \theta = 1 \][/tex]
Solving for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{64}{289} = \frac{289}{289} - \frac{64}{289} = \frac{225}{289} \][/tex]
So:
[tex]\[ \cos \theta = \pm \sqrt{\frac{225}{289}} = \pm \frac{15}{17} \][/tex]
### Summary:
Both verification conditions are checked and satisfied. Thus, both procedures and checks here illustrate the correctness of the solutions presented.
However, the conclusion drawn from the earlier steps shows:
[tex]\[ \boxed{4} \][/tex]
Therefore, based on the calculations and verifications:
Neither procedure is correct.
