find exact value of cos(12°) cos(33°) − sin(12°) sin(33°)

We can solve this expression using the **cosine addition formula**, which states:

\[
\cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)
\]

In your case, the given expression is:

\[
\cos(12^\circ) \cos(33^\circ) - \sin(12^\circ) \sin(33^\circ)
\]

This matches the form of the cosine addition formula with \(A = 12^\circ\) and \(B = 33^\circ\), so we can rewrite it as:

\[
\cos(12^\circ + 33^\circ) = \cos(45^\circ)
\]

The exact value of \(\cos(45^\circ)\) is:

\[
\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

Thus, the exact value of \(\cos(12^\circ) \cos(33^\circ) - \sin(12^\circ) \sin(33^\circ)\) is:

\[
\boxed{\frac{\sqrt{2}}{2}}
\]

Would you like more details on this solution, or have any questions?

Here are 5 related questions to explore further:
1. How do you derive the cosine addition formula?
2. What is the sine addition formula?
3. Can this formula be applied to angles greater than 90°?
4. How does the cosine subtraction formula differ from the addition formula?
5. How can you find the exact value of \(\sin(45^\circ)\)?

**Tip:** Memorizing key trigonometric identities like the addition and subtraction formulas can simplify many complex calculations.

Discover the exact value of cos(12°) cos(33°) − sin(12°) sin(33°) using the cosine addition formula. This detailed solution explains how to apply trigonometric identities to find the precise cosine of angles, suitable for high school students and beyond.

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