Trigonometric Double Angle Identities — Quiz

Question 7 • Precalculus

Which derivation correctly uses the cosine sum identity to prove the cosine double angle identity?

Answer

Table with 2 columns and 4 rows.Row 1: Column 1, empty; Column 2, 'Step' Row 2: Column 1, '1.'; Column 2, 'cosine of 2 x is equal to cosine of open paren x plus x close paren' Row 3: Column 1, '2.'; Column 2, 'is equal to sine x plus cosine x' Row 4: Column 1, '3.'; Column 2, 'is equal to sine squared of x plus cosine squared of x'

Table with 2 columns and 4 rows.Row 1: Column 1, empty; Column 2, 'Step' Row 2: Column 1, '1.'; Column 2, 'cosine of 2 x is equal to cosine of open paren x plus x close paren' Row 3: Column 1, '2.'; Column 2, 'is equal to cosine x plus sine x' Row 4: Column 1, '3.'; Column 2, 'is equal to cosine squared of x plus sine squared of x'

Table with 2 columns and 4 rows.Row 1: Column 1, empty; Column 2, 'Step' Row 2: Column 1, '1.'; Column 2, 'cosine of 2 x is equal to cosine of open paren x plus x close paren' Row 3: Column 1, '2.'; Column 2, 'is equal to cosine x minus sine x' Row 4: Column 1, '3.'; Column 2, 'is equal to cosine squared of x minus sine squared of x'

Table with 2 columns and 4 rows.Row 1: Column 1, empty; Column 2, 'Step' Row 2: Column 1, '1.'; Column 2, 'cosine of 2 x is equal to cosine of open paren x plus x close paren' Row 3: Column 1, '2.'; Column 2, 'is equal to sine x minus cosine x' Row 4: Column 1, '3.'; Column 2, 'is equal to sine squared of x minus cosine squared of x'

Explanation

Use the cosine sum identity: cos(a + b) = cos a cos b − sin a sin b. Set a = x and b = x, so cos(2x) = cos x cos x − sin x sin x = cos² x − sin² x. Choice C gives the correct final identity.

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