To solve for [tex]\(\tan \theta\)[/tex] given that [tex]\(\sec \theta + \tan \theta = 2\)[/tex], we'll use trigonometric identities and algebraic manipulation.
1. Let [tex]\(\sec \theta = x\)[/tex]. Then, it follows that:
[tex]\[
x + \tan \theta = 2
\][/tex]
Solving for [tex]\(\tan \theta\)[/tex], we get:
[tex]\[
\tan \theta = 2 - x
\][/tex]
2. Recall the trigonometric identity:
[tex]\[
\sec^2 \theta - \tan^2 \theta = 1
\][/tex]
Substituting [tex]\(\sec \theta = x\)[/tex] and [tex]\(\tan \theta = 2 - x\)[/tex], we have:
[tex]\[
x^2 - (2 - x)^2 = 1
\][/tex]
3. Expanding and simplifying the expression:
[tex]\[
x^2 - (4 - 4x + x^2) = 1
\][/tex]
[tex]\[
x^2 - 4 + 4x - x^2 = 1
\][/tex]
[tex]\[
4x - 4 = 1
\][/tex]
4. Solving for [tex]\(x\)[/tex]:
[tex]\[
4x - 4 = 1
\][/tex]
[tex]\[
4x = 5
\][/tex]
[tex]\[
x = \frac{5}{4}
\][/tex]
5. Substitute [tex]\(x\)[/tex] back into [tex]\(\tan \theta = 2 - x\)[/tex]:
[tex]\[
\tan \theta = 2 - \frac{5}{4} = \frac{8}{4} - \frac{5}{4} = \frac{3}{4}
\][/tex]
Therefore, the value of [tex]\(\tan \theta\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
