Evaluate

coscos

StartFraction 7 pi Over 12 EndFraction7π12

as

coscos

left parenthesis StartFraction pi Over 4 EndFraction plus StartFraction pi Over 3 EndFraction right parenthesisπ4+π3.

To evaluate \( \cos\left(\frac{7\pi}{12}\right) \) as \( \cos\left(\frac{\pi}{4} + \frac{\pi}{3}\right) \), we will use the **sum identity** for cosine:

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

In this case, \( A = \frac{\pi}{4} \) and \( B = \frac{\pi}{3} \). Now let's substitute these into the identity:

\[
\cos\left(\frac{\pi}{4} + \frac{\pi}{3}\right) = \cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{3}\right)
\]

We know the exact values of these trigonometric functions:

- \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
- \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)
- \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
- \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)

Now substituting these values:

\[
\cos\left(\frac{7\pi}{12}\right) = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}
\]

Simplifying:

\[
\cos\left(\frac{7\pi}{12}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}
\]

Factoring out \( \frac{1}{4} \):

\[
\cos\left(\frac{7\pi}{12}\right) = \frac{1}{4} \left(\sqrt{2} - \sqrt{6}\right)
\]

Thus, the value of \( \cos\left(\frac{7\pi}{12}\right) \) is:

\[
\boxed{\frac{1}{4} \left(\sqrt{2} - \sqrt{6}\right)}
\]

Would you like further explanation or details? Here are 5 related questions:

1. How do you derive the sum and difference identities for cosine?
2. Can you evaluate \( \sin\left(\frac{7\pi}{12}\right) \) using a similar approach?
3. How would you evaluate trigonometric expressions involving other angles like \( \frac{5\pi}{12} \)?
4. Can you prove the exact values of \( \cos\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{3}\right) \)?
5. How can you use sum identities to solve trigonometric equations?

**Tip:** Familiarize yourself with the common angles and their exact trigonometric values; they are frequently useful in many problems!

Evaluate cos(7π/12) as cos(π/4 + π/3).

Learn how to evaluate cos(7π/12) by applying the sum identity for cosine. This step-by-step solution involves trigonometric values for π/4 and π/3. Perfect for high school students studying trigonometry (Grades 10-12).

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