Sum to Product — Hard Practice Quiz

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

Practice quiz

1. Simplify the expression $\frac{\sin A + \sin B}{\cos A + \cos B}$.

   • $\tan\left(\frac{A + B}{2}\right)$
   • $\cot\left(\frac{A + B}{2}\right)$
   • $\tan\left(\frac{A - B}{2}\right)$
   • $\cot\left(\frac{A - B}{2}\right)$

   Answer: $\tan\left(\frac{A + B}{2}\right)$

2. If $\sin x + \sin y = 0$ and $\cos x + \cos y = 0$, which of the following must be true?

   • $x - y = (2n+1)\pi$ for some integer $n$
   • $x + y = (2n+1)\pi$ for some integer $n$
   • $x = y + 2n\pi$ for some integer $n$
   • $x = y$

   Answer: $x - y = (2n+1)\pi$ for some integer $n$

3. Simplify the expression $\frac{\sin 5x - \sin 3x}{\cos 5x + \cos 3x}$.

   • $\tan x$
   • $\cot x$
   • $\tan 4x$
   • $\cot 4x$

   Answer: $\tan x$

4. Solve for $x$ in the equation $\sin 3x + \sin x = \cos 3x + \cos x$.

   • $x = \frac{\pi}{2} + n\pi$ or $x = \frac{\pi}{8} + \frac{n\pi}{2}$ for integer $n$
   • $x = n\pi$ or $x = \frac{\pi}{4} + n\pi$ for integer $n$
   • $x = \frac{\pi}{4} + \frac{n\pi}{2}$ for integer $n$
   • $x = \frac{\pi}{2} + 2n\pi$ for integer $n$

   Answer: $x = \frac{\pi}{2} + n\pi$ or $x = \frac{\pi}{8} + \frac{n\pi}{2}$ for integer $n$

5. If $\alpha + \beta = \pi$, what is the value of $\cos \alpha + \cos \beta$?

   • $0$
   • $1$
   • $2\cos\left(\frac{\alpha - \beta}{2}\right)$
   • $-2\sin\left(\frac{\alpha - \beta}{2}\right)$

   Answer: $0$

6. Simplify the product $(\sin A + \sin B)(\cos A + \cos B)$.

   • $2\sin(A + B)\cos^2\left(\frac{A - B}{2}\right)$
   • $2\cos(A + B)\sin^2\left(\frac{A - B}{2}\right)$
   • $\sin(A + B)\cos(A - B)$
   • $4\sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)$

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

7. Using the sum-to-product identities, express $2\sin x \cos y$ as a sum or difference of sines.

   • $\sin(x+y) + \sin(x-y)$
   • $\sin(x+y) - \sin(x-y)$
   • $\cos(x+y) + \cos(x-y)$
   • $\cos(x+y) - \cos(x-y)$

   Answer: $\sin(x+y) + \sin(x-y)$

8. Simplify the expression $\frac{\sin A + \sin B}{\sin A - \sin B}$.

   • $\tan\left(\frac{A + B}{2}\right)\cot\left(\frac{A - B}{2}\right)$
   • $\cot\left(\frac{A + B}{2}\right)\tan\left(\frac{A - B}{2}\right)$
   • $\tan\left(\frac{A + B}{2}\right)\tan\left(\frac{A - B}{2}\right)$
   • $\cot\left(\frac{A + B}{2}\right)\cot\left(\frac{A - B}{2}\right)$

   Answer: $\tan\left(\frac{A + B}{2}\right)\cot\left(\frac{A - B}{2}\right)$

9. If $\sin(\theta + \phi) + \sin(\theta - \phi) = \cos(\theta + \phi) + \cos(\theta - \phi)$, and $\cos \phi \neq 0$, what is the value of $\tan \theta$?

   • $1$
   • $-1$
   • $0$
   • Undefined

   Answer: $1$

10. Given $\sin A + \sin B = X$ and $\cos A + \cos B = Y$. What is the value of $X^2 + Y^2$?

    • $4\cos^2\left(\frac{A - B}{2}\right)$
    • $4\sin^2\left(\frac{A - B}{2}\right)$
    • $4\cos^2\left(\frac{A + B}{2}\right)$
    • $4\sin^2\left(\frac{A + B}{2}\right)$

    Answer: $4\cos^2\left(\frac{A - B}{2}\right)$