[tex]\text{Given that,}\\\\[/tex]
[tex]~~~~~~~\cos \theta = \dfrac 89\\\\\\\implies 2\cos^2 \left(\dfrac{\theta}2 \right) -1 = \dfrac 89~~~~~~;\left[1+ \cos 2\theta = 2\cos^2 \theta \right]\\\\\\\implies 2 \cos^2 \left(\dfrac{\theta}2 \right)= \dfrac 89 +1\\\\\\\implies 2 \cos^2 \left(\dfrac{\theta}2 \right)=\dfrac{17}9\\\\\\\implies \cos^2 \left(\dfrac{\theta}2 \right) = \dfrac{17}{18}\\\\\\\implies \cos \left(\dfrac{\theta}2 \right)=\sqrt{\dfrac{17}{18}}~~~~~~~~~~~;\left[\text{Positive value of} ~\cos \left(\dfrac{\theta}2\right) \right][/tex]
[tex]\implies \cos \left(\dfrac{\theta} 2 \right) =\dfrac{\sqrt{17}}{\sqrt{9 \times 2}}\\\\\\\implies \cos \left(\dfrac{\theta} 2 \right)=\dfrac{\sqrt{17}}{3\sqrt 2}\\\\\\\implies \cos \left(\dfrac{\theta} 2 \right)=\dfrac{\sqrt{17} \cdot \sqrt 2}{3\cdot 2}\\\\\\\implies \cos \left(\dfrac{\theta} 2 \right)=\dfrac{\sqrt{34}}{6 }[/tex]
