To determine the value of [tex]\(\tan 45^\circ\)[/tex], let's analyze the given choices and see if we can match them correctly with the true value of [tex]\(\tan 45^\circ\)[/tex]:
We know that [tex]\(\tan 45^\circ\)[/tex] results in a numerical value close to 0.9999999999999999. Now, we will go through each of the given choices to check which one matches our result.
Choice A: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
[tex]\[ \frac{1}{\sqrt{2}} \approx 0.707 \][/tex]
This clearly does not match our result of approximately 0.9999999999999999.
Choice B: 1
[tex]\[ 1.0 \][/tex]
This is very close to our result of approximately 0.9999999999999999, but since it's not exactly equal, let's consider the precision. [tex]\(\tan 45^\circ\)[/tex] should ideally be exactly 1.
Choice C: [tex]\(\sqrt{2}\)[/tex]
[tex]\[ \sqrt{2} \approx 1.414 \][/tex]
This is also far from 0.9999999999999999.
Choice D: [tex]\(\frac{1}{2}\)[/tex]
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
This is also not close to 0.9999999999999999.
Since none of the other choices match as closely as Choice B, the closest value to 0.9999999999999999 is indeed 1. Therefore, we conclude that:
[tex]\(\tan 45^\circ = 1\)[/tex]
Hence, the correct option is:
B. 1
