Calcolare il valore della seguente espressione:
[math]((\\cos^2(18^circ))-(\\sin^2(18^circ)))/((\\cos^2(240^circ)))-(tg^2(-18^circ))[/math]
[math](\\cos^2(18^circ)-\\sin^2(18^circ))/(\\cos^2(240^circ))-tg^2(-18^circ)=>p>>/p> Essendo [/math]
cos(18^circ)=1/4sqrt(10+2sqrt5) , sin(18^circ)=1/4(sqrt5-1) , cos(240^circ)=-1/2 , tg(-18^circ)=-sqrt(1-2/5sqrt5)[math], sostituendo
ell'espressio
e si ha: [/math]
=((1/4sqrt(10+2sqrt5))^2-(1/4(sqrt5-1))^2)/((-1/2)^2)-(-sqrt(1-2/5sqrt5))^2=ell'espressio
e si ha: [/math]
[math]>p>>/p> [/math]
=(1/(16)(10+2sqrt5)-1/(16)(sqrt5-1)^2)/(1/4)-(1-2/5sqrt5)=[math]>p>>/p> [/math]
=(1/(16)(10+2sqrt5)-1/(16)(5+1-2sqrt5))/(1/4)-(1-2/5sqrt5)=[math]>p>>/p> [/math]
=(5/8+(sqrt5)/8-3/8+(sqrt5)/8)*4-1+2/5sqrt5=(1/4+(sqrt5)/4)*4-1+2/5sqrt5=[math]>p>>/p> [/math]
=1+sqrt5-1+2/5sqrt5=(5sqrt5+2sqrt5)/5=7/5sqrt5$.
