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  • If f(x)= cos^4(pi/8)+cos^4(3pi/8)+cos^4(5pi/8)+cos^4(7pi/8), and g(3/2)=1, then (gof)(x) equals

If f(x)= cos^4(pi/8)+cos^4(3pi/8)+cos^4(5pi/8)+cos^4(7pi/8), and g(3/2)=1, then (gof)(x) equals

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Solution:   We can write  ,

                    cos (frac5pi8)=cos(pi-frac3pi8)=-cos(frac3pi8)

               Similarly ,     cos (frac7pi8)=cos(pi-fracpi8)=-cos(fracpi8).

               herefore        f(x)=[2cos^4fracpi8+2cos^4frac3pi8]

                        f(x)=frac12[(1+cos fracpi4)^2+(1+cos frac3pi4)^2]\ \Rightarrow f(x)=frac12[(1+frac1sqrt2)^2+(1-frac1sqrt2)^2]=frac32.

                   Rightarrow      (gof)(x)=g[f(x)]=g(frac32)=1.         

Posted by

Deependra Verma

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