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Quizzes > Mathematics

Trigonometric Identities Quiz: Simplify, Prove, and Verify

Quick, free trig identities test. Instant results and answer checks.

Editorial: Review CompletedCreated By: Paul WoodwardUpdated Aug 27, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration for a trigonometry identities quiz on a golden yellow background.

This trigonometric identities quiz helps you practice simplifying, proving, and verifying identities across easy to hard items. Build recall with a trig identities memorization game, check your progress in a trigonometric identities unit test, and refresh angles on the unit circle quiz. Work at your own pace and spot gaps before a test.

What is the value of sin(?/6)?
?2/2
1
1/2
?3/2
On the unit circle, the coordinate for ?/6 (30°) has a y-value of 1/2. The sine of an angle corresponds to its y-coordinate on the unit circle. Therefore sin(?/6) equals 1/2. Learn more at .
What is the value of tan(45°)?
Undefined
1
?3
0
Tangent is defined as sin divided by cos. At 45° both sine and cosine equal ?2/2, so their ratio is 1. Hence tan(45°) = 1. See for more.
Which expression is the reciprocal identity for cos??
tan? = 1/cot?
csc? = 1/cos?
sec? = 1/cos?
cos? = 1/sec?
By definition, secant is the reciprocal of cosine, so sec? = 1/cos?. While cos? = 1/sec? is algebraically equivalent, the standard reciprocal identity is sec? = 1/cos?. More details at .
Which Pythagorean identity is correct?
sec²? - tan²? = tan²?
sin²? + cos²? = 1
tan²? + 1 = cot²?
1 + cot²? = tan²?
The fundamental Pythagorean identity is sin²? + cos²? = 1, derived from the unit circle equation x² + y² = 1. Other forms involve dividing by cos²? or sin²? to get sec²? and tan²? identities. For more, visit .
What is the result of tan? × cos??
tan²?
sin?
sec?
cos²?
tan? is sin?/cos?, so multiplying by cos? gives sin?. This cancellation is a straightforward application of the definition tan? = sin?/cos?. See for definitions.
What is the definition of cotangent (cot?)?
sin? / cos?
cos? / sin?
1 / tan?
tan? / 1
Cotangent is defined as the reciprocal of tangent, or cos? divided by sin?. While 1/tan? is equivalent, the primary definition is cot? = cos?/sin?. More at .
Evaluate sec²? ? tan²? for any ? where defined.
sec²?
1
0
tan²?
From the Pythagorean identity 1 + tan²? = sec²?, rearranging yields sec²? ? tan²? = 1. This holds for all ? where the functions are defined. Further explanation at .
What is the value of cos(?/2)?
Undefined
1
-1
0
On the unit circle, ?/2 (90°) corresponds to the point (0, 1). The cosine of an angle is its x-coordinate, which at ?/2 is 0. See for the full chart.
Which is the double-angle identity for sin(2?)?
2tan?/(1 + tan²?)
sin²? ? cos²?
1 ? 2sin²?
2sin?cos?
The double-angle formula for sine is sin(2?) = 2sin?cos?. This comes directly from the sum identity sin(a + b). See for derivation.
What is the formula for cos(a + b)?
cos a cos b ? sin a sin b
sin a sin b ? cos a cos b
sin a cos b ? cos a sin b
cos a cos b + sin a sin b
The sum of angles formula for cosine is cos(a + b) = cos a cos b ? sin a sin b. It follows from rotating vectors or using complex exponentials. More at .
Simplify tan(?/4 + ?) using the tangent addition formula.
1 ? tan?
tan? + 1
(tan? ? 1)/(tan? + 1)
(1 + tan?)/(1 ? tan?)
Using tan(a + b) = (tan a + tan b)/(1 ? tan a tan b) with a = ?/4 (tan a =1) gives (1 + tan?)/(1 ? tan?). More on .
Express sin x + sin y using the sum-to-product identity.
cos((x + y)/2)cos((x ? y)/2)
2cos((x + y)/2)sin((x + y)/2)
sin((x + y)/2)sin((x ? y)/2)
2sin((x + y)/2)cos((x ? y)/2)
The sum-to-product identity for sine states sin x + sin y = 2sin((x+y)/2)cos((x?y)/2). This is derived by applying sum and difference formulas. See .
Which is the power-reduction identity for cos²??
(1 ? cos2?)/2
(1 + cos2?)/2
2cos?
1 ? sin2?
Power-reduction formulas convert squares to first powers: cos²? = (1 + cos2?)/2, derived from the double-angle identity for cosine. Learn more at .
Which is the power-reduction identity for sin²??
(1 ? cos2?)/2
2sin?
1 ? cos?
(1 + cos2?)/2
Similarly, sin²? = (1 ? cos2?)/2 comes from rearranging cos2? = 1 ? 2sin²?. Using these simplifies integrals and proofs. See .
Expand (sin? + cos?)².
sin²? + 2sin?cos? + cos²?
sin²? + cos²?
sin²? ? 2sin?cos? + cos²?
2sin²?cos²?
Squaring a binomial yields a² + 2ab + b², so (sin? + cos?)² = sin²? + 2sin?cos? + cos²?. Recognizing Pythagorean identities can simplify further. More at .
Which identity represents sin A cos B + cos A sin B?
cos(A + B)
sin(A + B)
sin(A ? B)
cos(A ? B)
The sum formula for sine is sin(A + B) = sinA cosB + cosA sinB. This fundamental identity is used throughout trigonometry. See for more.
Which identity shows that 1 ? cos2? equals 2sin²??
1 + cos2? = 2sin²?
cos2? ? 1 = 2cos²?
1 ? 2cos²? = sin2?
1 ? cos2? = 2sin²?
From the double-angle formula cos2? = 1 ? 2sin²?, rearrange to get 1 ? cos2? = 2sin²?. This form is often used in integrals and proofs. Reference: .
Simplify tan? + cot? into a single expression.
1/(sin? cos?)
sin²? + cos²?
(sin? + cos?)/(sin? cos?)
(1 + tan²?)/tan?
tan? + cot? = sin?/cos? + cos?/sin? = (sin²? + cos²?)/(sin? cos?) = 1/(sin? cos?) since sin²? + cos²? = 1. More at .
Which identity expresses cos x ? cos y as a product?
2 cos((x + y)/2) sin((x ? y)/2)
-2 sin((x + y)/2) sin((x ? y)/2)
2 sin((x + y)/2) cos((x ? y)/2)
2 cos((x ? y)/2) sin((x ? y)/2)
The difference-to-product formula states cos x ? cos y = ?2 sin((x+y)/2) sin((x?y)/2). This identity is useful for simplifying expressions and integrals. See .
Which of the following is a correct product-to-sum identity for sin x cos y?
sin(x + y)sin(x ? y)
½[cos(x + y) ? cos(x ? y)]
½[sin(x + y) + sin(x ? y)]
½[sin(x + y) ? sin(x ? y)]
The product-to-sum identity for sin x cos y is sin x cos y = ½[sin(x+y) + sin(x?y)]. It's derived using sum and difference formulas. More details at .
Which identity relates sin? to tan(?/2)?
sin? = (1 + tan²(?/2))/2tan(?/2)
sin? = 2tan(?/2)/(1 + tan²(?/2))
sin? = (1 ? tan²(?/2))/(1 + tan²(?/2))
sin? = tan(?/2)
The half-angle to full-angle conversion gives sin? = 2tan(?/2)/(1 + tan²(?/2)). This identity simplifies integration and complex proofs. See .
Which identity relates cos? to tan(?/2)?
cos? = (1 ? tan²(?/2))/(1 + tan²(?/2))
cos? = tan(?/2)
cos? = (1 + tan²(?/2))/(1 ? tan²(?/2))
cos? = 2tan(?/2)/(1 + tan²(?/2))
From half-angle formulas, cos? = (1 ? tan²(?/2))/(1 + tan²(?/2)). This relation is useful in integration and series expansions. More at .
What is the triple-angle identity for sin(3?)?
4sin³? ? 3sin?
sin?(3 ? sin²?)
3sin? + 4sin³?
3sin? ? 4sin³?
The triple-angle formula for sine is sin(3?) = 3sin? ? 4sin³?, derived from sum identities and double-angle formulas. It often appears in Fourier analysis. Reference: .
Which trigonometric identity represents tan A + tan B?
sin(A + B)/(cos A cos B)
(tan A tan B)/(1 ? tan A tan B)
(cos A + cos B)/(sin A sin B)
sin(A ? B)/(sin A cos B)
Starting with tanA + tanB = (sinA/cosA) + (sinB/cosB), combine fractions to get (sinA cosB + cosA sinB)/(cosA cosB), which simplifies to sin(A+B)/(cosA cosB). See .
Simplify (1 + sin?)/cos? into half-angle functions.
1 + tan?
tan(?/2) + sec(?/2)
sec? + tan?
tan? + 1/cos?
Using half-angle identities, 1 + sin? = sin²(?/2) + cos²(?/2) + 2sin(?/2)cos(?/2) = (sin(?/2) + cos(?/2))² and cos? = cos²(?/2) ? sin²(?/2). Simplifying yields tan(?/2) + sec(?/2). More at .
0
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Study Outcomes

  1. Understand fundamental trigonometric identities -

    Recognize and recall basic Pythagorean, reciprocal, and quotient identities to build a solid foundation for advanced trig proofs.

  2. Apply identities to simplify expressions -

    Rewrite complex trigonometric expressions in simpler forms using strategic identity substitutions for more efficient problem-solving.

  3. Prove trigonometric identities -

    Develop step-by-step approaches to transform one side of an equation into the other, reinforcing logical reasoning and proof-writing skills.

  4. Analyze angle relationships -

    By tackling quiz 6-1 basic trigonometric identities questions, explore how cofunction and negative-angle identities reveal connections between sine, cosine, and other ratios.

  5. Solve basic trigonometric equations -

    Use proven identities to isolate variables and determine exact values of trigonometric functions within standard angle measures.

  6. Evaluate trigonometric expressions -

    Compute numeric values of trig expressions at key angles confidently by leveraging simplified identities for accurate and quick results.

Cheat Sheet

  1. Pythagorean Identities -

    The foundation of most trig identities, like sin²θ + cos²θ = 1, comes from the unit circle. According to Khan Academy and MIT OpenCourseWare, you can derive 1 + tan²θ = sec²θ and cot²θ + 1 = csc²θ by dividing through by cos²θ or sin²θ. Remembering "sin squared plus cos squared equals one" is key for many trig identities quiz questions.

  2. Reciprocal and Quotient Identities -

    Reciprocal identities define secθ = 1/cosθ, cscθ = 1/sinθ, and cotθ = 1/tanθ, while quotient identities express tanθ = sinθ/cosθ and cotθ = cosθ/sinθ. These are fundamental for simplifying complex trig expressions in a trigonometric identities quiz according to Paul's Online Math Notes. A simple mnemonic is "Shaco, Cahsoh, Toa," to recall sine over cosine, cosine over sine, and tangent relationships.

  3. Angle Sum and Difference Identities -

    The formulas sin(a±b) = sin a cos b ± cos a sin b and cos(a±b) = cos a cos b ∓ sin a sin b are vital for trig identities questions involving compound angles. MIT OpenCourseWare encourages practicing with examples like sin 75° by rewriting it as sin(45°+30°) for efficient computation. A handy trick is remembering that the sign in the sine formula matches the original operation while cosine flips the sign.

  4. Double-Angle and Half-Angle Identities -

    Double-angle identities such as sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ - sin²θ (or 2 cos²θ - 1 or 1 - 2 sin²θ) are staples in our trig identities quiz practice. From these, you can derive half-angle formulas like sin²(θ/2) = (1 - cos θ)/2, which is crucial for integration and proving identities as noted by Stewart's Calculus. These identities often simplify expressions in quiz 6-1 basic trigonometric identities/proving trigonometric identities tasks.

  5. Proving Identities: Techniques and Strategies -

    Successful proofs often start by converting everything to sine and cosine, working on one side of the equation, and looking for like terms to factor or cancel. University of Texas's online resources recommend checking both sides by choosing strategic angles to verify before diving into algebraic steps. Keeping these strategies in mind boosts your confidence when facing challenging trig identities quiz questions.

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