Question

Sa se demonstreze ca urmatoarele expresii nu depind de x:

a) E = cos^2(x - x^2) + cos^2(x^2) - 2cos(x - x^2)cos(x)cos(x^2) + cos^3x
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Answer 1

Răspuns:

$cos^2(x-x^2)+cos^2(x^2)-2cosx*cos(x^2)*cos(x-x^2)+cos^2x$

$cos^2(x-x^2)+cos^2(x^2)-(cos(x+x^2)+cos(x-x^2))*cos(x-x^2)+cos^2x$

(Obs: Am transformat $2cos(x)*cos(x^2)=cos(x+x^2)+cos(x-x^2)$)

$cos^2(x-x^2)+cos^2(x^2)-cos^2(x-x^2)-cos(x-x^2)*cos(x+x^2)+cos^2x$

$cos^2(x^2)+cos^2x-cos(x-x^2)*cos(x+x^2)$

$cos^2(x^2)+cos^2x-\frac{cos(x-x^2+x+x^2)+cos(x-x^2-x-x^2)}{2}$

(Obs: Am transformat produsul in suma)

$\frac{2cos^2(x^2)+2cos^2x-cos(2x)-cos(2x^2)}{2}$

(Obs: Am adus la acelasi numitor)

$\frac{2cos^2(x^2)+2cos^2x-cos^2x+sin^2x-cos^2(x^2)+sin^2(x^2)}{2}$

$\frac{sin^2x+cos^2x+sin^2(x^2)+cos^2(x^2)}{2}$

$\frac{1+1}{2}=\frac{2}{2}=1$ => Expresia nu depinde de x