Let's solve the equation [tex]\(\cos x = \sin (20^\circ + x)\)[/tex] given the constraint [tex]\(0^\circ < x < 90^\circ\)[/tex].
1. Recall the trigonometric identity:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
2. Applying this identity to the given equation:
[tex]\[
\cos x = \sin (20^\circ + x)
\][/tex]
can be rewritten using the identity as:
[tex]\[
\cos x = \cos(90^\circ - (20^\circ + x))
\][/tex]
3. Simplify the expression inside the cosine:
[tex]\[
\cos x = \cos(90^\circ - 20^\circ - x)
\][/tex]
[tex]\[
\cos x = \cos(70^\circ - x)
\][/tex]
4. Since cosine is an even function and [tex]\( \cos A = \cos B \)[/tex] implies that [tex]\( A = B \)[/tex], we have:
[tex]\[
x = 70^\circ - x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x + x = 70^\circ
\][/tex]
[tex]\[
2x = 70^\circ
\][/tex]
[tex]\[
x = \frac{70^\circ}{2}
\][/tex]
[tex]\[
x = 35^\circ
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\( \boxed{35} \)[/tex].
