The value of x in (-pi ,pi) which satisfies the equation 8^(1+|cosx| +|cos^2 x| +|cos^3 x|+........to infinite ) =4^3 is

Answers (1)

Solution:    Let   S=1+left | cos x 
ight |+left | cos^2x 
ight |+left | cos^3x 
ight |+........infty

                 It is an infinite G.P    whose common ratio

                                       r=left | cos x 
ight |< 1

                  	herefore               S=fraca1-r=frac11-left | cos x 
ight |

               Hence , the given equation reduces  to

                                8^(frac11-left | cos x 
ight |)=4^3Rightarrow 2^(frac31-left | cos x 
ight |)=2^6

             Rightarrow             frac31-left | cos x 
ight |=6Rightarrow frac1-left | cos x 
ight |3=frac16

               Rightarrow                  1-left | cos x 
ight |=frac12 

            	herefore                left | cos x 
ight |=frac12Rightarrow x=fracpi3,-fracpi3.

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