Addition and Subtraction — Hard Practice Quiz

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

Formulas & key concepts

Sine Sum

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

Sine Difference

$$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$

Cosine Sum

$$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$

Cosine Difference

$$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$

Tangent Sum

$$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$

Tangent Difference

$$\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$$

Practice quiz

  1. Which of the following expressions is equivalent to $\sin(2\alpha)$?

    • $\frac{2 \tan \alpha}{1 + \tan^2 \alpha}$
    • $\sin^2 \alpha - \cos^2 \alpha$
    • $\frac{1 - \tan^2 \alpha}{1 + \tan^2 \alpha}$
    • $\cos^2 \alpha - \sin^2 \alpha$

    Answer: $\frac{2 \tan \alpha}{1 + \tan^2 \alpha}$

  2. Given $\sin \alpha = \frac{3}{5}$ where $\alpha$ is in Quadrant II, and $\cos \beta = \frac{12}{13}$ where $\beta$ is in Quadrant IV, find the exact value of $\tan(\alpha + \beta)$.

    • $-\frac{56}{33}$
    • $\frac{56}{33}$
    • $\frac{33}{56}$
    • $-\frac{33}{56}$

    Answer: $-\frac{56}{33}$

  3. Simplify the expression $\cos(x + y) \cos y + \sin(x + y) \sin y$.

    • $\cos x$
    • $\sin x$
    • $\cos(x + 2y)$
    • $\sin(x + 2y)$

    Answer: $\cos x$

  4. Evaluate the exact value of $\frac{\tan 75^\circ - \tan 15^\circ}{1 + \tan 75^\circ \tan 15^\circ}$.

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

    Answer: $\sqrt{3}$

  5. If $\alpha + \beta = \frac{\pi}{4}$, which of the following statements is true?

    • $(1 + \tan \alpha)(1 + \tan \beta) = 2$
    • $(1 - \tan \alpha)(1 - \tan \beta) = 2$
    • $\tan \alpha + \tan \beta = 1$
    • $\tan \alpha \tan \beta = 1$

    Answer: $(1 + \tan \alpha)(1 + \tan \beta) = 2$

  6. Which of the following expressions is equivalent to $\cot(\alpha + \beta)$?

    • $\frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}$
    • $\frac{\cot \alpha \cot \beta + 1}{\cot \alpha + \cot \beta}$
    • $\frac{1 - \cot \alpha \cot \beta}{\cot \alpha + \cot \beta}$
    • $\frac{\cot \alpha + \cot \beta}{1 - \cot \alpha \cot \beta}$

    Answer: $\frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}$

  7. If $\sin x \cos 30^\circ + \cos x \sin 30^\circ = \frac{1}{2}$, what is the general solution for $x$?

    • $x = 2\pi k$ or $x = \frac{2\pi}{3} + 2\pi k$
    • $x = \frac{\pi}{6} + 2\pi k$ or $x = \frac{5\pi}{6} + 2\pi k$
    • $x = \frac{\pi}{3} + 2\pi k$ or $x = \frac{2\pi}{3} + 2\pi k$
    • $x = \frac{\pi}{2} + 2\pi k$ or $x = \frac{3\pi}{2} + 2\pi k$

    Answer: $x = 2\pi k$ or $x = \frac{2\pi}{3} + 2\pi k$

  8. If $\alpha$ and $\beta$ are acute angles such that $\sin \alpha = \frac{1}{\sqrt{5}}$ and $\sin \beta = \frac{1}{\sqrt{10}}$, what is $\alpha + \beta$?

    • $\frac{\pi}{4}$
    • $\frac{\pi}{2}$
    • $\frac{\pi}{3}$
    • $\frac{\pi}{6}$

    Answer: $\frac{\pi}{4}$

  9. Simplify the expression $\frac{\sin(x + y)}{\cos x \cos y}$.

    • $\tan x + \tan y$
    • $\tan x \tan y$
    • $\cot x + \cot y$
    • $\cot x \cot y$

    Answer: $\tan x + \tan y$

  10. If $\tan \alpha = 2$ and $\tan \beta = 3$, what is the value of $\tan(\alpha - \beta)$?

    • $-\frac{1}{7}$
    • $\frac{1}{7}$
    • $-1$
    • $\frac{1}{5}$

    Answer: $-\frac{1}{7}$

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