$(4sin(60^\circ+x)sin(60^\circ-x))/(3sin^2(180^\circ+x))-1/(tg^2(180^\circ-x))$

A cura di: Francesco Speciale

Semplificare la seguente espressione
$(4sin(60^circ+x)sin(60^circ-x))/(3sin^2(180^circ+x))-1/(tg^2(180^circ-x))$

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$(4sin(60^circ+x)sin(60^circ-x))/(3sin^2(180^circ+x))-1/(tg^2(180^circ-x))$
Tenendo presente le formule di somma e differenza del seno e della tangente:

$sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)$
$sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)$
$tg(alpha-beta)=(tg(alpha)-tg(beta))/(1+tg(beta)tg(alpha))$

Riferendoci alla nosta espressione avremo:
$(4sin(60^circ+x)sin(60^circ-x))/(3sin^2(180^circ+x))-1/(tg^2(180^circ-x))=$
$(4(sin(60^circ)cos(x)+cos(60^circ)sin(x))(sin(60^circ)cos(x)-cos(60^circ)sin(x)))/(3(sin(180^circ)cos(x)+cos(180^circ)sin(x))^2)-1/(((tg(180^circ)-tgx)/(1-tg(180^circ)tgx))^2)=$
Essendo $cos(60^circ)=1/2, sin(60^circ)=(sqrt3)/2, cos(180^circ)=-1, sin(180^circ)=tg(180^circ)=0$,
sostituendo nell’espressione otteniamo:
$(4((sqrt3)/2cosx+1/2sinx)((sqrt3)/2cosx-1/2sinx))/(3((-1)sin(x))^2)-1/((-tgx)^2)=$
$(4(3/4cos^2x-(sqrt3)/4sinxcosx+(sqrt3)/4sinxcosx-1/4sin^2x))/(3sin^2x)-1/(tg^2x)=$
$(3cos^2x-sin^2x)/(3sin^2x)-cotg^2x=(3cos^2x)/(3sin^2x)-(sin^2x)/(3(sin^2x))-cotg^2x=$
Semplificando
$cotg^2x-1/3-cotg^2x=-1/3$.