Write another term using the cosine ratio that is equivalent to cos 75°.

To find another term that is equivalent to \( \cos 75^\circ \), we can use the angle subtraction identity for cosine:

\[
\cos(75^\circ) = \cos(45^\circ + 30^\circ)
\]

Using the cosine addition formula:

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

Substitute \( A = 45^\circ \) and \( B = 30^\circ \):

\[
\cos(75^\circ) = \cos(45^\circ) \cos(30^\circ) - \sin(45^\circ) \sin(30^\circ)
\]

Now, substitute the known values:

- \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \)
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)

So,

\[
\cos(75^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}
\]

Simplify each term:

\[
= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}
\]

Combine terms:

\[
= \frac{\sqrt{6} - \sqrt{2}}{4}
\]

Thus, another expression for \( \cos 75^\circ \) is:

\[
\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

Would you like a detailed explanation of each step or have any questions?

---

1. How can we use the sine ratio to find an equivalent for \( \sin 75^\circ \)?
2. What is the value of \( \cos 15^\circ \) using similar methods?
3. How would you express \( \cos 105^\circ \) in terms of known angles?
4. How can we verify \( \cos 75^\circ \) using a calculator and compare it to \( \frac{\sqrt{6} - \sqrt{2}}{4} \)?
5. Can the angle subtraction identity be used to find another equivalent form of \( \cos 75^\circ \)?

**Tip:** Remember, the angle addition and subtraction formulas are powerful tools for finding equivalent expressions of trigonometric functions for uncommon angles.

Learn how to express cos 75° in a different form using the cosine addition formula. This solution breaks down the steps using known angle values and trigonometric identities. Suitable for high school students in Grades 10-12 studying trigonometry.

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