cos(x+y)cos(x-y)=cos(^2)x-sin(^2)y

Calculus
Tutor: None Selected Time limit: 1 Day

Prove the cofunction identity using the Addition and Subtraction Formulas.

May 15th, 2015

cos(x+y)= cos(x)cos(y)-sin(x)sin(y)

cos(x-y)=cos(x)cos(y)+sin(x)sin(y)

then:

cos(x+y)*cos(x-y)=[cos(x)cos(y)-sin(x)sin(y)]*[cos(x)cos(y)+sin(x)sin(y)]

Using the identity: a^2-b^2=(a+b)*(a-b), we get:

=[cos(x)cos(y)]^2-[sin(x)sin(y)]^2

and using [cos(z)]^2+[sin(z)]^2 = 1
we have

=[cos(x)^2] * [1-(sin(y))^2] - [sin(x)]^2 * [1-(cos(y))^2]

The mixed terms cancel so we finally have:

=cos(x)^2 - (sin(y))^2

as desired

Good Luck !!


May 15th, 2015

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