Pythagorean Identities — Hard Practice Quiz

A Trigonometry cheat sheet for Pythagorean Identities — every key formula with its symbols defined — plus a hard-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. Simplify the expression: $(\sec \alpha - \tan \alpha)(\sec \alpha + \tan \alpha)$.

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

    Answer: $1$

  2. If $\sin \alpha = \frac{3}{5}$ and $\alpha$ is an acute angle, what is the value of $\tan \alpha$?

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

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

  3. Express $\frac{1 - \sin^2 \alpha}{1 - \cos^2 \alpha}$ in terms of $\cot \alpha$.

    • $\cot^2 \alpha$
    • $\tan^2 \alpha$
    • $\sec^2 \alpha$
    • $\csc^2 \alpha$

    Answer: $\cot^2 \alpha$

  4. Simplify the expression: $\frac{\sin \alpha}{1 + \cos \alpha} + \frac{1 + \cos \alpha}{\sin \alpha}$.

    • $2 \csc \alpha$
    • $2 \sec \alpha$
    • $2 \tan \alpha$
    • $2 \cot \alpha$

    Answer: $2 \csc \alpha$

  5. If $\sec \alpha = x$, express $\sin \alpha$ in terms of $x$ (assume $\sin \alpha > 0$).

    • $\frac{\sqrt{x^2 - 1}}{x}$
    • $\frac{x}{\sqrt{x^2 - 1}}$
    • $\frac{1}{x}$
    • $\frac{x^2 - 1}{x}$

    Answer: $\frac{\sqrt{x^2 - 1}}{x}$

  6. Given $\tan \alpha = k$, express $\csc^2 \alpha$ in terms of $k$.

    • $\frac{k^2 + 1}{k^2}$
    • $\frac{k^2}{k^2 + 1}$
    • $k^2 + 1$
    • $\frac{1}{k^2 + 1}$

    Answer: $\frac{k^2 + 1}{k^2}$

  7. Simplify the expression: $(\sin \alpha + \cos \alpha)^2 + (\sin \alpha - \cos \alpha)^2$.

    • $2$
    • $1$
    • $2 \sin^2 \alpha$
    • $2 \cos^2 \alpha$

    Answer: $2$

  8. If $\sin \alpha + \cos \alpha = \frac{1}{2}$, what is the value of $\sin \alpha \cos \alpha$?

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

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

  9. Express $\frac{\sec \alpha - \cos \alpha}{\tan \alpha}$ in terms of $\sin \alpha$.

    • $\sin \alpha$
    • $\cos \alpha$
    • $\tan \alpha$
    • $\cot \alpha$

    Answer: $\sin \alpha$

  10. If $\tan \alpha = \frac{p}{q}$, express $\frac{\sin \alpha - \cos \alpha}{\sin \alpha + \cos \alpha}$ in terms of $p$ and $q$.

    • $\frac{p - q}{p + q}$
    • $\frac{p + q}{p - q}$
    • $\frac{p^2 - q^2}{p^2 + q^2}$
    • $\frac{p^2 + q^2}{p^2 - q^2}$

    Answer: $\frac{p - q}{p + q}$

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