Question #326597

 Show that the functions 1, cos2 x, sin2 x are linearly dependent .


1
Expert's answer
2022-04-11T16:38:12-0400

Assume that there are complex numbers a1,a2,a3a_1,a_2,a_3 not all equal to zero satisfying: a1+a2cos2x+a3sin2x=0a_1+a_2cos\,2x+a_3sin\,2x=0. Choose different values of x:x: x=0,x=π4,x=π3x=0,x=\frac{\pi}{4},x=\frac{\pi}{3}. We get the following equations: a1+a2=0,a1+a3=0=0a112a2+a332=0a_1+a_2=0,\,\, a_1+a_3=0=0\,\,a_1-\frac12a_2+a_3\frac{\sqrt{3}}{2}=0. From first and second equations we get: a2=a3=a1a_2=a_3=-a_1 . Third equation implies: a3=0a_3=0. Thus, all numbers are zero. We came to contradiction and received that all functions are linearly independent.


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