To solve for [tex]\(x\)[/tex] in the equation [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], where [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a trigonometric identity to transform the equation into a simpler form. The key identity we will use is:
[tex]\[
\sin(90^\circ - \theta) = \cos(\theta)
\][/tex]
Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], this identity suggests that:
[tex]\[
\cos(x) = \sin(90^\circ - x)
\][/tex]
For our specific case, this means:
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
By the identity [tex]\(\sin(90^\circ - \theta) = \cos(\theta)\)[/tex], we can equate:
[tex]\[
\sin(14^\circ) = \cos(76^\circ)
\][/tex]
Thus,
[tex]\[
\cos(x) = \cos(76^\circ)
\][/tex]
Since the cosine function is unique in the interval [tex]\(0^\circ < x < 90^\circ\)[/tex], comparing both sides, we have:
[tex]\[
x = 76^\circ
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] is:
[tex]\[
\boxed{76^\circ}
\][/tex]
