To evaluate the expression

$\left( \frac{\cos 75^\circ + \cos 15^\circ}{\sin 75^\circ - \sin 15^\circ} \right)^2$

without using a calculator, we can use trigonometric identities for sum and difference of cosines and sines. Let’s solve it step by step:

1. Simplify the numerator: $\cos 75^\circ + \cos 15^\circ$

Use the sum-to-product identity for cosines: $\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$

Here, $A = 75^\circ$ and $B = 15^\circ$. So: $\cos 75^\circ + \cos 15^\circ = 2 \cos\left(\frac{75^\circ + 15^\circ}{2}\right) \cos\left(\frac{75^\circ - 15^\circ}{2}\right)$ $= 2 \cos(45^\circ) \cos(30^\circ)$

Now, substitute the exact values: $\cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}$ $\cos 75^\circ + \cos 15^\circ = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$

2. Simplify the denominator: $\sin 75^\circ - \sin 15^\circ$

Use the difference-to-product identity for sines: $\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$

Here, $A = 75^\circ$ and $B = 15^\circ$. So: $\sin 75^\circ - \sin 15^\circ = 2 \cos\left(\frac{75^\circ + 15^\circ}{2}\right) \sin\left(\frac{75^\circ - 15^\circ}{2}\right)$ $= 2 \cos(45^\circ) \sin(30^\circ)$

Now, substitute the exact values: $\cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2}$ $\sin 75^\circ - \sin 15^\circ = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}$

3. Form the fraction:

The fraction becomes: $\frac{\cos 75^\circ + \cos 15^\circ}{\sin 75^\circ - \sin 15^\circ} = \frac{\frac{\sqrt{6}}{2}}{\frac{\sqrt{2}}{2}} = \frac{\sqrt{6}}{\sqrt{2}}$

Simplify: $\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3}$

4. Square the result:

Finally, square the fraction: $\left(\frac{\cos 75^\circ + \cos 15^\circ}{\sin 75^\circ - \sin 15^\circ}\right)^2 = (\sqrt{3})^2 = 3$

Final Answer:

$\boxed{3}$

Would you like further clarification or practice on similar problems? Here are related questions:

1. Can you derive the sum-to-product and difference-to-product identities?
2. What are the exact values of common trigonometric functions like $\cos 15^\circ$ or $\sin 75^\circ$?
3. How would you approach this problem geometrically?
4. Could you verify this result using another trigonometric identity?
5. How does this problem relate to the unit circle?

Tip: Always double-check your exact values for common angles when solving without a calculator!