Product to Sum — Hard Practice Quiz

A Trigonometry cheat sheet for Product to Sum — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. Simplify the expression $4 \sin(3x) \cos(2x) \sin(x)$.

   • $1 + \cos(4x) - \cos(2x) - \cos(6x)$
   • $1 - \cos(2x) + \cos(4x) - \cos(6x)$
   • $2 \sin(x) \cos(5x) + 2 \sin^2(x)$
   • $\cos(x) - \cos(3x) + \cos(5x) - \cos(7x)$

   Answer: $1 - \cos(2x) + \cos(4x) - \cos(6x)$

2. Express $\cos(7x) + \cos(3x)$ as a product of trigonometric functions.

   • $2 \cos(5x) \cos(2x)$
   • $2 \sin(5x) \sin(2x)$
   • $2 \cos(5x) \sin(2x)$
   • $2 \sin(5x) \cos(2x)$

   Answer: $2 \cos(5x) \cos(2x)$

3. If $\sin(5x) \cos(3x) = \frac{1}{2} \sin(8x) + \frac{1}{4}$, find the general solution for $x$, where $n$ is an integer.

   • $x = \frac{\pi}{12} + n\pi$ or $x = \frac{5\pi}{12} + n\pi$
   • $x = \frac{\pi}{6} + n\pi$ or $x = \frac{5\pi}{6} + n\pi$
   • $x = \frac{\pi}{24} + n\frac{\pi}{2}$ or $x = \frac{5\pi}{24} + n\frac{\pi}{2}$
   • $x = \frac{\pi}{4} + n\pi$

   Answer: $x = \frac{\pi}{12} + n\pi$ or $x = \frac{5\pi}{12} + n\pi$

4. Consider the identity $\cos \alpha \cdot \cos \beta = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)]$. If $\alpha = \beta$, which fundamental identity does this simplify to?

   • $\cos^2 \alpha = \frac{1 + \cos(2\alpha)}{2}$
   • $\sin^2 \alpha = \frac{1 - \cos(2\alpha)}{2}$
   • $\cos(2\alpha) = \cos^2 \alpha - \sin^2 \alpha$
   • $\sin(2\alpha) = 2 \sin \alpha \cos \alpha$

   Answer: $\cos^2 \alpha = \frac{1 + \cos(2\alpha)}{2}$

5. Evaluate the expression $\sin(\frac{\pi}{12}) \sin(\frac{5\pi}{12}) \cos(\frac{\pi}{6})$.

   • $\frac{\sqrt{3}}{8}$
   • $\frac{1}{8}$
   • $\frac{\sqrt{3}}{4}$
   • $\frac{1}{4}$

   Answer: $\frac{\sqrt{3}}{8}$

6. Using the given product-to-sum identities, derive an expression for $\sin A - \sin B$.

   • $2 \cos(\frac{A+B}{2}) \sin(\frac{A-B}{2})$
   • $2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$
   • $-2 \cos(\frac{A+B}{2}) \sin(\frac{A-B}{2})$
   • $2 \sin(\frac{A-B}{2}) \cos(\frac{A+B}{2})$

   Answer: $2 \cos(\frac{A+B}{2}) \sin(\frac{A-B}{2})$

7. Evaluate $\cos(75^\circ) \cos(15^\circ) - \sin(75^\circ) \sin(15^\circ)$.

   • $0$
   • $\frac{1}{2}$
   • $\frac{\sqrt{3}}{2}$
   • $1$

   Answer: $0$

8. If $\sin(A) \cos(B) = \frac{1}{2}$ and $\sin(A) \sin(B) = \frac{\sqrt{3}}{2}$, what is the value of $\cos(A+B)$?

   • $-\frac{\sqrt{3}}{2}$
   • $\frac{1}{2}$
   • $\frac{\sqrt{3}}{2}$
   • $-1$

   Answer: $-\frac{\sqrt{3}}{2}$

9. Simplify the expression $\cos(x) \cos(2x) - \sin(3x) \sin(x)$.

   • $\frac{1}{2}[\cos(x) + \cos(3x) - \cos(2x) + \cos(4x)]$
   • $\frac{1}{2}[\cos(x) + \cos(3x) + \cos(2x) - \cos(4x)]$
   • $\frac{1}{2}[\cos(x) - \cos(3x) - \cos(2x) + \cos(4x)]$
   • $\frac{1}{2}[\cos(x) - \cos(3x) + \cos(2x) - \cos(4x)]$

   Answer: $\frac{1}{2}[\cos(x) + \cos(3x) - \cos(2x) + \cos(4x)]$

10. If $\sin(A) \cos(B) = k$ and $\cos(A) \sin(B) = m$, what is the value of $\sin(A+B) \sin(A-B)$ in terms of $k$ and $m$?

    • $k^2 - m^2$
    • $k^2 + m^2$
    • $(k+m)^2$
    • $(k-m)^2$

    Answer: $k^2 - m^2$