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  • 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

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

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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.

Posted by

Deependra Verma

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