Problem 2 Deriving a Double-Angle Identity

How can you derive the identity cos 2 θ = 1 − 2 sin 2 θ ? cosine 2 theta equals 1 minus 2 , sine squared , theta question mark

Use the Pythagorean identity sin 2 θ + cos 2 θ = 1 sine squared , theta plus , cosine squared , theta equals 1 to get cos 2 θ = 1 − sin 2 θ . cosine squared , theta equals 1 minus , sine squared , theta .

cos 2 θ = cos 2 θ − sin 2 θ Cosine Double-Angle Identity   = ( 1 − sin 2 θ ) − sin 2 θ Use the Pythagorean identity .   = 1 − 2 sin 2 θ Simplify . table with 3 rows and 3 columns , row1 column 1 , cosine 2 theta , column 2 equals , cosine squared , theta minus , sine squared , theta , column 3 cap cosinecap doubleminuscap anglecap identity , row2 column 1 , , column 2 equals . open . 1 minus , sine squared , theta . close . minus , sine squared , theta , column 3 cap usethecap pythagoreanidentity . . , row3 column 1 , , column 2 equals 1 minus 2 , sine squared , theta , column 3 cap simplify , . , end table

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1. How can you derive the identity cos 2 θ = 2 cos 2 θ − 1 ? cosine 2 theta equals 2 , cosine squared , theta negative 1 question mark

Problem 2 Using a Double-Angle Identity

What is the exact value of cos 120 ° ? cosine , 120 degrees question mark Use a double-angle identity.

cos 120 ° = cos ( 2 · 60 ° ) Rewrite  120  as  ( 2 · 60 ) .   = cos 2 60 ° − sin 2 60 ° Cosine Double-Angle Identity   = ( 1 2 ) 2 − ( 3 2 ) 2 Replace with exact values .   = − 1 2 Simplify . table with 4 rows and 3 columns , row1 column 1 , cosine , 120 degrees , column 2 equals cosine . open . 2 middle dot 60 degrees . close , column 3 cap rewrite . 120 , as , open 2 middle dot 60 close . , row2 column 1 , , column 2 equals , cosine squared , 60 degrees negative , sine squared , 60 degrees , column 3 cap cosinecap doubleminuscap anglecap identity , row3 column 1 , , column 2 equals . open , 1 half , close squared . minus . open , fraction square root of 3 , over 2 end fraction , close squared , column 3 cap replacewithexactvalues . . , row4 column 1 , , column 2 equals negative , 1 half , column 3 cap simplify , . , end table

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2. What is the exact value of sin 120 ° ? sine , 120 degrees question mark Use a double-angle identity.

Problem 3 Verifying an Identity

Verify the identity cos 2 θ = 1 − tan 2 θ 1 + tan 2 θ . cosine 2 theta equals . fraction 1 minus , tangent squared , theta , over 1 plus , tangent squared , theta end fraction . .

1 − tan 2 θ 1 + tan 2 θ = 1 − tan 2 θ sec 2 θ Use a Pythagorean identity .   = 1 sec 2 θ − tan 2 θ sec 2 θ Write as two fractions .   = 1 1 cos 2 θ − sin 2 θ cos 2 θ 1 cos 2 θ Express in terms of  sin θ  and  cos θ .   = cos 2 θ − sin 2 θ Simplify .   = cos 2 θ Cosine Double-Angle Identity table with 5 rows and 3 columns , row1 column 1 , fraction 1 minus , tangent squared , theta , over 1 plus , tangent squared , theta end fraction , column 2 equals . fraction 1 minus , tangent squared , theta , over secant squared , theta end fraction , column 3 cap useacap pythagoreanidentity . . , row2 column 1 , , column 2 equals . fraction 1 , over secant squared , theta end fraction . minus . fraction tangent squared , theta , over secant squared , theta end fraction , column 3 cap writeastwofractions . . , row3 column 1 , , column 2 equals . fraction 1 , over fraction 1 , over cosine squared , theta end fraction end fraction . minus . fraction fraction sine squared , theta , over cosine squared , theta end fraction , over fraction 1 , over cosine squared , theta end fraction end fraction , column 3 cap expressintermsof sine theta , and cosine theta . , row4 column 1 , , column 2 equals , cosine squared , theta minus , sine squared , theta , column 3 cap simplify , . , row5 column 1 , , column 2 equals cosine 2 theta , column 3 cap cosinecap doubleminuscap anglecap identity , end table