We can solve $\cos^2 75^\circ - \sin^2 75^\circ$ using the trigonometric identity:

$\cos^2 x - \sin^2 x = \cos(2x)$

Step-by-step solution:

1. Here, $x = 75^\circ$.
2. Substitute $x = 75^\circ$ into the identity: $\cos^2 75^\circ - \sin^2 75^\circ = \cos(2 \cdot 75^\circ)$
3. Simplify $2 \cdot 75^\circ$: $\cos(150^\circ)$
4. Using the unit circle, $\cos(150^\circ) = -\frac{\sqrt{3}}{2}$.

Final Answer:

$\cos^2 75^\circ - \sin^2 75^\circ = -\frac{\sqrt{3}}{2}$

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Would you like a detailed explanation of the trigonometric identity or a breakdown of cosine values for specific angles? Let me know!

5 Related Questions:

1. What is the derivation of the formula $\cos^2 x - \sin^2 x = \cos(2x)$?
2. How do you calculate $\sin(150^\circ)$ or $\tan(150^\circ)$?
3. Can this formula be used to simplify other angles, such as $\cos^2 45^\circ - \sin^2 45^\circ$?
4. What is the value of $\cos 75^\circ$ and $\sin 75^\circ$ individually?
5. How do trigonometric identities help in simplifying equations?

Tip:

Always remember the unit circle properties to quickly find trigonometric values for standard angles like $30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ,$ and more.