To find the value of [tex]\( x \)[/tex] for which [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] in the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use the co-function identity for trigonometric functions:
[tex]\[
\cos(x) = \sin(90^\circ - x)
\][/tex]
Given the equation [tex]\( \cos(x) = \sin(14^\circ) \)[/tex], we can equate the expressions:
[tex]\[
\cos(x) = \sin(14^\circ)
\][/tex]
By the co-function identity, we know:
[tex]\[
\cos(x) = \sin(90^\circ - x)
\][/tex]
Therefore, we can write:
[tex]\[
\sin(90^\circ - x) = \sin(14^\circ)
\][/tex]
Since the sine function is positive and strictly increasing in the range [tex]\( 0^\circ \)[/tex] to [tex]\( 90^\circ \)[/tex], the equality [tex]\( \sin(90^\circ - x) = \sin(14^\circ) \)[/tex] implies:
[tex]\[
90^\circ - x = 14^\circ
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
90^\circ - x = 14^\circ
\][/tex]
[tex]\[
x = 90^\circ - 14^\circ
\][/tex]
[tex]\[
x = 76^\circ
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 76^\circ \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{76^\circ}
\][/tex]
