Express Sin 40 Sin 30 as a sum or difference and also Cos 60° - Cos 20° as a product.

Let's express \(\sin 40^\circ \sin 30^\circ\) as a sum or difference and \(\cos 60^\circ - \cos 20^\circ\) as a product.

### 1. Expressing \(\sin 40^\circ \sin 30^\circ\) as a sum or difference:

We use the product-to-sum identity for sine functions:

\[
\sin A \sin B = \frac{1}{2} \left[\cos(A-B) - \cos(A+B)\right]
\]

Here, \(A = 40^\circ\) and \(B = 30^\circ\):

\[
\sin 40^\circ \sin 30^\circ = \frac{1}{2} \left[\cos(40^\circ - 30^\circ) - \cos(40^\circ + 30^\circ)\right]
\]

\[
= \frac{1}{2} \left[\cos 10^\circ - \cos 70^\circ\right]
\]

So, \(\sin 40^\circ \sin 30^\circ = \frac{1}{2} (\cos 10^\circ - \cos 70^\circ)\).

### 2. Expressing \(\cos 60^\circ - \cos 20^\circ\) as a product:

We use the sum-to-product identity for cosine functions:

\[
\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)
\]

Here, \(A = 60^\circ\) and \(B = 20^\circ\):

\[
\cos 60^\circ - \cos 20^\circ = -2 \sin\left(\frac{60^\circ + 20^\circ}{2}\right) \sin\left(\frac{60^\circ - 20^\circ}{2}\right)
\]

\[
= -2 \sin(40^\circ) \sin(20^\circ)
\]

So, \(\cos 60^\circ - \cos 20^\circ = -2 \sin 40^\circ \sin 20^\circ\).

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Would you like more details or have any questions about these expressions?

Here are some related questions you might find interesting:
1. How do you express \(\cos 40^\circ \cos 30^\circ\) as a sum or difference?
2. What is the product-to-sum identity for \(\sin A \cos B\)?
3. Can these identities be used in solving trigonometric equations?
4. How would you apply these identities to verify trigonometric identities?
5. What are the sum-to-product identities for sine functions?

**Tip:** Memorizing these identities can make solving trigonometric problems much faster and easier!

Learn how to express \(\sin 40^\circ \sin 30^\circ\) as a sum or difference using trigonometric identities, and find \(\cos 60^\circ - \cos 20^\circ\) as a product. This problem involves using key trigonometric identities and is suitable for high school students.

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