The exact value of cos 112.5 is [tex]\bold{-\frac{\sqrt{(2-\sqrt{2})}}{2}= 0.3827531}[/tex]
Solution:
Use half angle formula for Cos,
[tex]\begin{array}{l}{\cos \left(\frac{x}{2}\right)=\pm \sqrt{\frac{1+\cos x}{2}}} \\ {\cos 112.5=\cos \frac{225}{2}=-\sqrt{\frac{1+\cos 225}{2}} \quad(\text { equation } 1)}\end{array}[/tex]
(Since cos 112.5 is in II quadrant ,negative sign is used)
cos 225 = cos (45+180)
cos (a+b) = cos a cos b+sin a sin b
cos (45+180) = cos 45 cos 180+ sin 45 sin 180
[tex]\begin{array}{l}{\cos 45=\frac{\sqrt{2}}{2}} \\ \\ {\sin 45=\frac{\sqrt{2}}{2}}\end{array}[/tex]
cos 180 = -1
sin 180 = 0
[tex]\begin{array}{l}{\cos (45+180)=\frac{\sqrt{2}}{2}(-1)+\frac{\sqrt{2}}{2}(0)} \\\\ {\cos (45+180)=-\frac{\sqrt{2}}{2}+0} \\\\ {\cos (45+180)=-\frac{\sqrt{2}}{2}(\text { equation } 2)}\end{array}[/tex]
apply equation 2 in equation 1
[tex]\begin{array}{l}{\cos \frac{225}{2}=-\sqrt{\frac{1+\left(-\frac{\sqrt{2}}{2}\right)}{2}}=-\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}=-\sqrt{\frac{2-\sqrt{2}}{2}}=-\sqrt{\frac{2-\sqrt{2}}{4}}} \\ {\cos \frac{225}{2}=-\frac{\sqrt{2-\sqrt{2}}}{2}} \\ {\cos 112.5=-\frac{\sqrt{2-\sqrt{2}}}{2}}=0.3827531\end{array}[/tex]
