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Find the sum of all possible values of x satisfying arc `cos((2)/(pi) arc cos x)=arc sin((2)/(pi) arc sinx)`. |
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Answer» 1 arc `cos((2)/(pi) arc cos x)=arc sin ((2)/(pi) arc sin x) ` `cos^(-1)((2)/(pi) ((pi)/(2)-sin^(-1)x))=sin^(-1)((2)/(pi) sin^(-1)x)` `cos^(-1)(1-(2)/(pi) sin^(-1)x)=sin^(-1)((2)/(pi) sin^(-1)x)` Let `(2)/(pi) sin^(-1)x=alpha` were `alpha in [0,1]` think ! `implies cos^(-1)(1-alpha)=sin^(-1) alpha` `implies sin^(-1) sqrt(2alpha-alpha^(2))=sin^(-1) alpha implies sqrt(2alpha-alpha^(2))=alphaimplies 2 alpha-alpha^(2)=alpha^(2)` `implies 2 alpha=2alpha^(2)` Hence `alpha` is either 0 or 1 If `alpha=0` then x=0 if `alpha=1` then x=1 hence sum of all possible value of x is 1 |
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