Question: write as sum or differnece $\cos x \sin 7x$
1. **State the problem:** We want to express the product $\cos x \sin 7x$ as a sum or difference of trigonometric functions.
2. **Recall the product-to-sum formula:**
$$\cos A \sin B = \frac{1}{2} [\sin(A+B) - \sin(A-B)]$$
3. **Apply the formula with $A = x$ and $B = 7x$:**
$$\cos x \sin 7x = \frac{1}{2} [\sin(x + 7x) - \sin(x - 7x)]$$
4. **Simplify inside the sine functions:**
$$= \frac{1}{2} [\sin 8x - \sin(-6x)]$$
5. **Use the odd property of sine:**
$$\sin(-\theta) = -\sin \theta$$
So,
$$\sin(-6x) = -\sin 6x$$
6. **Substitute back:**
$$\cos x \sin 7x = \frac{1}{2} [\sin 8x - (-\sin 6x)] = \frac{1}{2} [\sin 8x + \sin 6x]$$
**Final answer:**
$$\cos x \sin 7x = \frac{1}{2} (\sin 8x + \sin 6x)$$