alleryStart learning →How is the sum formula for tangent obtained?
ABy taking the product of the sum formulas for sine and cosine
BBy adding the sum formulas for sine and cosine term by term
CBy taking the quotient of the sum formulas for sine and cosine
DBy squaring the sum formula for sine and halving the result
Answer & Solution
Correct answer: C. By taking the quotient of the sum formulas for sine and cosine
1. Tangent is sine over cosine, so tan(a + b) = sin(a + b) / cos(a + b).
2. Substituting the sum formulas gives (sin a cos b + cos a sin b) / (cos a cos b - sin a sin b).
3. Dividing numerator and denominator by cos a cos b turns every term into a tangent.
4. The result is (tan a + tan b) / (1 - tan a tan b), so the route is a quotient of the sine and cosine sum formulas.
5. Adding the two formulas term by term mixes sines and cosines and never produces a tangent.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.2 Sum and Difference Identities_
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