Respuesta:
Resolvemos:
[tex]\sin \left75^{\circ \:}\right\cos \left30^{\circ \:}\right-\sin \left30^{\circ \:}\right\cdot\cos \left75^{\circ \:}\right=\sin 75^{\circ \:}-30^{\circ \:}[/tex]
[tex]\sin45^{\circ \:}=\frac{\sqrt{2}}{2}[/tex]
[tex]\cos \left60^{\circ \:}\right\cdot\cos \left15^{\circ \:}\right+\sin \left15^{\circ \:}\right\cdot\sin \left60^{\circ \:}\right=\cos 60^{\circ \:}-15^{\circ \:}[/tex]
[tex]\cos 45^{\circ \:}=\frac{\sqrt{2}}{2}[/tex]
[tex]\sin \left75^{\circ \:}\right\cos \left30^{\circ \:}\right-\sin \left30^{\circ \:}\right\cdot\cos \left75^{\circ \:}\right=\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}-\frac{1}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}[/tex]
[tex]\cos \left60^{\circ \:}\right\cdot\cos \left15^{\circ \:}\right+\sin \left15^{\circ \:}\right\cdot\sin \left60^{\circ \:}\right=\frac{1}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}+\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}[/tex]
[tex]\boxed{\mathrm{E=\frac{\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}-\frac{1}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}}{\frac{1}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}+\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}}}}[/tex]
Simplificamos:
[tex]\mathrm{Multiplicar \ por \ el \ conjugado:}[/tex]
[tex]\frac{\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}}[/tex]
[tex]\frac{\left(\sqrt{3}\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)}{\left(\sqrt{2+\sqrt{3}}+\sqrt{3}\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)}[/tex]
[tex]\left(\sqrt{3}\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)=4\sqrt{3}-4[/tex]
[tex]\left(\sqrt{2+\sqrt{3}}+\sqrt{3}\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)=4\sqrt{3}-4[/tex]
[tex]\frac{4\sqrt{3}-4}{4\sqrt{3}-4}=1[/tex]
Solución: Respuesta correcta (Opción A)
[tex]\boxed{\mathrm{E=1}}[/tex]
Saludos...
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Respuesta:
CALCULAR:
[tex]\boxed{\mathrm{E=\frac{\sin \left75^{\circ \:}\right\cos \left30^{\circ \:}\right-\sin \left30^{\circ \:}\right\cdot\cos \left75^{\circ \:}\right}{\cos \left60^{\circ \:}\right\cdot\cos \left15^{\circ \:}\right+\sin \left15^{\circ \:}\right\cdot\sin \left60^{\circ \:}\right}}}[/tex]
Resolvemos:
[tex]\sin \left75^{\circ \:}\right\cos \left30^{\circ \:}\right-\sin \left30^{\circ \:}\right\cdot\cos \left75^{\circ \:}\right=\sin 75^{\circ \:}-30^{\circ \:}[/tex]
[tex]\sin45^{\circ \:}=\frac{\sqrt{2}}{2}[/tex]
[tex]\cos \left60^{\circ \:}\right\cdot\cos \left15^{\circ \:}\right+\sin \left15^{\circ \:}\right\cdot\sin \left60^{\circ \:}\right=\cos 60^{\circ \:}-15^{\circ \:}[/tex]
[tex]\cos 45^{\circ \:}=\frac{\sqrt{2}}{2}[/tex]
[tex]\sin \left75^{\circ \:}\right\cos \left30^{\circ \:}\right-\sin \left30^{\circ \:}\right\cdot\cos \left75^{\circ \:}\right=\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}-\frac{1}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}[/tex]
[tex]\cos \left60^{\circ \:}\right\cdot\cos \left15^{\circ \:}\right+\sin \left15^{\circ \:}\right\cdot\sin \left60^{\circ \:}\right=\frac{1}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}+\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}[/tex]
[tex]\boxed{\mathrm{E=\frac{\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}-\frac{1}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}}{\frac{1}{2}\cdot \frac{\sqrt{2+\sqrt{3}}}{2}+\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2-\sqrt{3}}}{2}}}}[/tex]
Simplificamos:
[tex]\mathrm{Multiplicar \ por \ el \ conjugado:}[/tex]
[tex]\frac{\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}}[/tex]
[tex]\frac{\left(\sqrt{3}\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)}{\left(\sqrt{2+\sqrt{3}}+\sqrt{3}\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)}[/tex]
[tex]\left(\sqrt{3}\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)=4\sqrt{3}-4[/tex]
[tex]\left(\sqrt{2+\sqrt{3}}+\sqrt{3}\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}-\sqrt{3}\sqrt{2-\sqrt{3}}\right)=4\sqrt{3}-4[/tex]
[tex]\frac{4\sqrt{3}-4}{4\sqrt{3}-4}=1[/tex]
Solución: Respuesta correcta (Opción A)
[tex]\boxed{\mathrm{E=1}}[/tex]
Saludos...