Calcule $$ Q=\frac{ an ^{4} 60^{\circ}-\sec ^{4} 45^{\circ}}{\left(\sec 30^{\circ}-\cot 60^{\circ} ight)^{2}} $$

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Calcule $$ Q=\frac{	an ^{4} 60^{\circ}-\sec ^{4} 45^{\circ}}{\left(\sec 30^{\circ}-\cot 60^{\circ}ight)^{2}} $$

UpStudy AI · Accepted Answer

Para calcular $$ Q $$:

$$
Q = \frac{\tan^{4} 60^{\circ} - \sec^{4} 45^{\circ}}{(\sec 30^{\circ} - \cot 60^{\circ})^{2}}
$$

Primero, calculamos cada una de las funciones trigonométricas:

1. $$\tan 60^{\circ} = \sqrt{3}$$, entonces $$\tan^{4} 60^{\circ} = (\sqrt{3})^{4} = 9$$.
2. $$\sec 45^{\circ} = \sqrt{2}$$, entonces $$\sec^{4} 45^{\circ} = (\sqrt{2})^{4} = 4$$.
3. $$\sec 30^{\circ} = \frac{2\sqrt{3}}{3}$$.
4. $$\cot 60^{\circ} = \frac{1}{\sqrt{3}}$$.

Sustituyendo estos valores en la expresión de $$ Q $$:

$$
Q = \frac{9 - 4}{\left(\frac{2\sqrt{3}}{3} - \frac{1}{\sqrt{3}}\right)^{2}} = \frac{5}{\left(\frac{\sqrt{3}}{3}\right)^{2}} = \frac{5}{\frac{1}{3}} = 15
$$

**El valor de $$ Q $$ es 15.**
El valor de $$ Q $$ es 15.

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