Factorised using the factor formulae, $\cos\theta-\cos3\theta-\cos5\theta+\cos7\theta$ equals:

1. Group the terms so each pair is a sum: $(\cos7\theta+\cos\theta)-(\cos5\theta+\cos3\theta)$. 2. Apply $\cos A+\cos B=2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$ to both pairs. 3. This gives $2\cos4\theta\cos3\theta-2\cos4\theta\cos\theta$. 4. Factor out $2\cos4\theta$: $2\cos4\theta(\cos3\theta-\cos\theta)$. 5. Now $\cos3\theta-\cos\theta=-2\sin2\theta\sin\theta$, so the product is $-4\cos4\theta\sin2\theta\sin\theta$. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/2, Question 5: Trigonometry_