To solve the equation [tex]\( 9 \cos^{-1}(x) - \pi = 3 \cos^{-1}(x) \)[/tex], follow these steps:
1. Let's set [tex]\( y = \cos^{-1}(x) \)[/tex]. This transforms the given equation into:
[tex]\[
9y - \pi = 3y
\][/tex]
2. Simplify the equation:
[tex]\[
9y - 3y = \pi
\][/tex]
[tex]\[
6y = \pi
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{\pi}{6}
\][/tex]
4. Given that [tex]\( y = \cos^{-1}(x) \)[/tex], we now have:
[tex]\[
\cos^{-1}(x) = \frac{\pi}{6}
\][/tex]
5. To solve for [tex]\( x \)[/tex], we take the cosine of both sides of the equation:
[tex]\[
x = \cos\left(\frac{\pi}{6}\right)
\][/tex]
6. Recall the value of [tex]\(\cos\left(\frac{\pi}{6}\right)\)[/tex]:
[tex]\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, the solution set is:
[tex]\[
\boxed{\left\{\frac{\sqrt{3}}{2}\right\}}
\][/tex]
