Cos^(2) ((1)/(2)*arccos(-(12)/(13) ))

Question

`cos^(2) ((1)/(2)*arccos(-(12)/(13) ))`

Answer 1

Recall the half-angle identity,

cos²(x/2) = (1 + cos(x))/2

This means

`\cos^2\left(\frac12 \arccos\left(-(12)/(13)\right)\right) = \frac{1 + \cos\left(\arccos\left(-(12)/(13)\right)\right)}2 = \frac{1 - (12)/(13)}2 = \boxed{\frac1{26}}`