Simplify sec^2θ - tan^2θ. [1 point]

Write cos(2θ) using only sinθ. [1 point]

College Board AP 2024 (style)-style practice: Section II (free response, no calculator). Given sinθ = (3)/(5) with θ in Quadrant I. (a) Find cosθ. (b) Use a double-angle identity to find sin(2θ).

A 3-point question on identities. (a) Cosine (1 point): cos^2θ = 1 - (9)/(25) = (16)/(25), and Quadrant I makes cosine positive, so cosθ = (4)/(5). (b) Double angle (2 points): sin(2θ) = 2sinθcosθ = 2 · (3)/(5) · (4)/(5) = (24)/(25).

United StatesPrecalculusSyllabus dot point

How do trigonometric identities let you rewrite an expression in an equivalent form?

Topic 3.12 Equivalent Representations of Trigonometric Functions: use the Pythagorean, sum and difference, and double-angle identities to rewrite trigonometric expressions in equivalent forms.

A focused answer to AP Precalculus Topic 3.12, covering the Pythagorean identities, the sum and difference formulas, and the double-angle formulas, and how to use them to rewrite trigonometric expressions and verify identities.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

Trigonometric identities are equations true for every angle, so they let you replace one expression with an equivalent one. The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ (and its forms $\tan^2\theta + 1 = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$ ) converts between sine, cosine and their reciprocals. The sum and difference formulas, such as $\sin(A + B) = \sin A\cos B + \cos A\sin B$ and $\cos(A + B) = \cos A\cos B - \sin A\sin B$ , break a combined angle into known pieces. The double-angle formulas, $\sin(2\theta) = 2\sin\theta\cos\theta$ and $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$ , follow from the sum formulas with $A = B = \theta$ . You use these to simplify expressions, evaluate non-standard angles, and verify that two expressions are equal.

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What this topic is asking
The Pythagorean identities
Sum and difference formulas
Double-angle formulas
Verifying identities
Try this

What this topic is asking

The College Board (Topic 3.12) wants you to rewrite trigonometric expressions in equivalent forms using identities: the Pythagorean identities, the sum and difference formulas, and the double-angle formulas. Recognizing that two different-looking expressions are equal, and converting one to the other, is the core skill, used to simplify, to evaluate, and to verify.

The Pythagorean identities

These are the workhorses for collapsing a mixed expression to a single function, and for swapping between a squared sine and a squared cosine.

Sum and difference formulas

These let you evaluate an angle by splitting it into two known angles, for example $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$ .

Double-angle formulas

Evaluating a non-standard angle

Find $\cos\frac{\pi}{12}$ exactly using a difference formula.

step 1 Write the angle as a difference of known angles

$\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ (that is $60^\circ - 45^\circ = 15^\circ$ ).

step 2 Apply the cosine difference formula

$\cos(A - B) = \cos A\cos B + \sin A\sin B$ , with $A = \frac{\pi}{3}$ , $B = \frac{\pi}{4}$ .

step 3 Substitute known values

$\cos\frac{\pi}{3} = \frac{1}{2}$ , $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$ , $\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ . So

\cos\frac{\pi}{12} = \frac{1}{2}\cdot\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}.

step 4 Combine

$\cos\frac{\pi}{12} = \frac{\sqrt{2} + \sqrt{6}}{4}$ , an exact value obtained without a calculator.

Verifying identities

To verify that two expressions are equal, work on the more complicated side and rewrite it using identities until it matches the other side. The strategy is to express everything in sines and cosines, apply the Pythagorean identity to merge terms, and simplify. You manipulate only one side at a time; you do not move terms across the equals sign as if solving an equation, because you are demonstrating the two sides are already equal.

A point worth stating once is that an identity is true for all angles, whereas an equation (Topic 3.10) is true only for specific ones. The two tasks look similar but call for opposite moves: verifying an identity means rewriting until both sides match, while solving an equation means isolating the variable. Keeping the goals distinct, "show they are always equal" versus "find where they are equal", prevents the common mistake of "solving" an identity and losing the structure of the proof.

Try this

Q1. Simplify $\sec^2\theta - \tan^2\theta$ . [1 point]

Cue. From $\tan^2\theta + 1 = \sec^2\theta$ , the difference is $1$ .

Q2. Write $\cos(2\theta)$ using only $\sin\theta$ . [1 point]

Cue. $\cos(2\theta) = 1 - 2\sin^2\theta$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part A (multiple choice, no calculator). Which expression is equivalent to

1 - \sin^2\theta

? (A)

\cos^2\theta

(B)

\tan^2\theta

(C)

2\cos\theta

(D)

\sec^2\theta

Show worked answer →

The correct answer is (A), $\cos^2\theta$ .

The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ rearranges to $1 - \sin^2\theta = \cos^2\theta$ . This is the most common rewrite on the exam, used to collapse a mixed expression to a single function.

AP 2024 (style)3 marksSection II (free response, no calculator). Given

\sin\theta = \frac{3}{5}

with

\theta

in Quadrant I. (a) Find

\cos\theta

. (b) Use a double-angle identity to find

\sin(2\theta)

Show worked answer →

A 3-point question on identities.

(a) Cosine (1 point): $\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$ , and Quadrant I makes cosine positive, so $\cos\theta = \frac{4}{5}$ .
(b) Double angle (2 points): $\sin(2\theta) = 2\sin\theta\cos\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$ .

Related dot points

Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)