To find the value of [tex]\( x \)[/tex] given the equation [tex]\(\cos x = \sin (20 + x)^{\circ}\)[/tex] for [tex]\(0^{\circ} < x < 90^{\circ}\)[/tex]:
1. Recall the co-function identity for trigonometric functions: [tex]\(\cos \theta = \sin (90^{\circ} - \theta)\)[/tex].
2. Use this identity to rewrite the given equation in terms of a sine function:
[tex]\[
\cos x = \sin (90^{\circ} - x)
\][/tex]
3. Therefore, we can equate:
[tex]\[
\sin (90^{\circ} - x) = \sin (20^{\circ} + x)
\][/tex]
4. Since the sine function is periodic, the general solution of [tex]\(\sin A = \sin B\)[/tex] is given by:
[tex]\[
90^{\circ} - x = 20^{\circ} + x
\][/tex]
(Or the other general solutions involving [tex]\(\pi\)[/tex] periods, which are not relevant here because we work in degrees between 0 and 90).
5. Solve the equation:
[tex]\[
90^{\circ} - x = 20^{\circ} + x
\][/tex]
6. Combine like terms:
[tex]\[
90^{\circ} - 20^{\circ} = x + x
\][/tex]
7. Simplify:
[tex]\[
70^{\circ} = 2x
\][/tex]
8. Divide both sides by 2:
[tex]\[
x = \frac{70}{2} = 35^{\circ}
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{35} \)[/tex].
