Question 138823
let 


*[Tex \LARGE u=\frac{1}{2}\cos^{-1}\left(-\frac{3}{4}\right) ]



so 


*[Tex \LARGE \cos\left(u\right)=\cos\left(\frac{1}{2}\cos^{-1}\left(-\frac{3}{4}\right)\right)]



*[Tex \LARGE \cos\left(2u\right)=\cos\left(2*\frac{1}{2}\cos^{-1}\left(-\frac{3}{4}\right)\right)] Substitute "2u" in for "u"



*[Tex \LARGE \cos\left(2u\right)=\cos\left(\cos^{-1}\left(-\frac{3}{4}\right)\right)] Multiply



*[Tex \LARGE \cos\left(2u\right)=-\frac{3}{4}] Take the arccosine of the cosine of {{{-3/4}}} to get {{{-3/4}}}



*[Tex \LARGE 2\cos^2\left(u\right)-1=-\frac{3}{4}] Replace  *[Tex \LARGE \cos\left(2u\right)] with *[Tex \LARGE 2\cos^2\left(u\right)-1]



*[Tex \LARGE 2\cos^2\left(u\right)=\frac{1}{4}] Add 1 to both sides



*[Tex \LARGE \cos^2\left(u\right)=\frac{1}{8}] Divide both sides by 2



*[Tex \LARGE \cos\left(u\right)=\sqrt{\frac{1}{8}}] Take the square root of both sides



*[Tex \LARGE \cos\left(u\right)=\frac{1}{\sqrt{8}}] Simplify



*[Tex \LARGE \cos\left(u\right)=\frac{\sqrt{2}}{4}] Rationalize the denominator if necessary




So *[Tex \LARGE \cos\left(\frac{1}{2}\cos^{-1}\left(-\frac{3}{4}\right)\right)=\frac{\sqrt{2}}{4}]