Pythagorean Identities — Practice Quiz

A Trigonometry cheat sheet for Pythagorean Identities — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Fundamental Pythagorean Identity

$$\sin^2 \alpha + \cos^2 \alpha = 1$$

Tangent-Secant Identity

$$\tan^2 \alpha + 1 = \sec^2 \alpha$$

Cotangent-Cosecant Identity

$$\cot^2 \alpha + 1 = \csc^2 \alpha$$

Quotient Identities

$$\tan \alpha = \frac{\sin \alpha}{\cos \alpha}, \quad \cot \alpha = \frac{\cos \alpha}{\sin \alpha}$$

Reciprocal Identity for Tangent and Cotangent

$$\tan \alpha \cdot \cot \alpha = 1$$

Reciprocal Identities

$$\sec \alpha = \frac{1}{\cos \alpha}, \quad \csc \alpha = \frac{1}{\sin \alpha}$$

Practice quiz

  1. If $\sin \alpha = 0.6$, what is $\cos^2 \alpha$?

    • $0.36$
    • $0.64$
    • $0.8$
    • $0.4$

    Answer: $0.64$

  2. Simplify the expression $\sec^2 \theta - \tan^2 \theta$.

    • $\sin^2 \theta$
    • $\cos^2 \theta$
    • $1$
    • $\tan \theta$

    Answer: $1$

  3. Which of the following is equivalent to $\csc^2 \beta - 1$?

    • $\tan^2 \beta$
    • $\cot^2 \beta$
    • $\sec^2 \beta$
    • $\sin^2 \beta$

    Answer: $\cot^2 \beta$

  4. If $\sin x = \frac{3}{5}$ and $\cos x = \frac{4}{5}$, what is $\tan x$?

    • $\frac{4}{3}$
    • $\frac{3}{4}$
    • $\frac{5}{3}$
    • $\frac{5}{4}$

    Answer: $\frac{3}{4}$

  5. What is the reciprocal of $\cos \theta$?

    • $\sin \theta$
    • $\tan \theta$
    • $\sec \theta$
    • $\csc \theta$

    Answer: $\sec \theta$

  6. Simplify $\sin \alpha \cdot \cot \alpha$.

    • $\tan \alpha$
    • $\cos \alpha$
    • $\sec \alpha$
    • $\csc \alpha$

    Answer: $\cos \alpha$

  7. If $\tan \theta = 2$, what is $\cot \theta$?

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

    Answer: $\frac{1}{2}$

  8. Which of the following identities is INCORRECT?

    • $\sin^2 \alpha + \cos^2 \alpha = 1$
    • $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$
    • $\sec \alpha = \frac{1}{\sin \alpha}$
    • $\cot^2 \alpha + 1 = \csc^2 \alpha$

    Answer: $\sec \alpha = \frac{1}{\sin \alpha}$

  9. Simplify $\frac{1 - \cos^2 x}{\sin x}$.

    • $\cos x$
    • $\sin x$
    • $\tan x$
    • $1$

    Answer: $\sin x$

  10. If $\cos A = \frac{1}{3}$, what is $\sec A$?

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

    Answer: $3$

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