Cosine Difference Identity
cos(A-B)=cosAcosB+sinAsinB
The cosine difference identity, also known as the cosine subtraction formula, is a trigonometric identity that expresses the cosine of the difference of two angles in terms of the product of the cosines of each angle and the sine of the difference of these angles.
The identity is given as follows:
cos(x – y) = cos(x)cos(y) + sin(x)sin(y)
where x and y are angles in radians.
This identity can be derived using the basic trigonometric identities of sine, cosine, and tangent. To derive the cosine difference identity, we start with the trigonometric identity:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
Next, we substitute a = x and b = -y into this identity to get:
sin(x – y) = sin(x)cos(-y) + cos(x)sin(-y)
Since cosine is an even function and sine is an odd function, we can simplify this expression as follows:
sin(x – y) = sin(x)cos(y) – cos(x)sin(y)
Finally, we divide both sides of the equation by sin(x) to get:
cos(x – y) = cos(x)cos(y) – sin(x)sin(y)
We can then use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to obtain the final form of the cosine difference identity:
cos(x – y) = cos(x)cos(y) + sin(x)sin(y)
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