Is the set of functions linearly dependent on the interval (-2,2) .pdf

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shankaranilahuja

Is the set of functions linearly dependent on the interval (-/2,/2) ?? the set is : cos2(x), sin2(x), sec2(x), tan2(x) Solution let acos^2x+b sin^2x +c sec^2x +d tan^2x =0 at x=0 a+c=0 at x=pi/6 3a/4+b/4+4/3c+d/3=0 at x=pi/4 a/2+b/2+2c+d=0 at x=pi/3 a/4+3b/4+4c+3d=0 this is only possible if a=0 b=0 c=0 d=0 therefore they are independent.

Is the set of functions linearly dependent on the interval (-/2,/2) ??
the set is : cos2(x), sin2(x), sec2(x), tan2(x)
Solution
let acos^2x+b sin^2x +c sec^2x +d tan^2x =0 at x=0 a+c=0 at x=pi/6
3a/4+b/4+4/3c+d/3=0 at x=pi/4 a/2+b/2+2c+d=0 at x=pi/3 a/4+3b/4+4c+3d=0 this is only
possible if a=0 b=0 c=0 d=0 therefore they are independent

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