Cosine Sum Identity
cos(A+B)=cosAcosB-sinAsinB
The cosine sum identity is a mathematical formula that expresses the sum of two cosine functions in terms of a single cosine function. It states that:
cos(a)cos(b) = [cos(a + b) + cos(a – b)]/2
This formula is useful in trigonometry and calculus, as it allows for simplification of trigonometric expressions involving multiple cosine terms.
To prove the cosine sum identity, we use the following trigonometric identities:
1) cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
2) cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
Now, adding equations (1) and (2) gives:
cos(a + b) + cos(a – b) = 2cos(a)cos(b)
Rearranging the terms and dividing by 2 gives us the desired result:
cos(a)cos(b) = [cos(a + b) + cos(a – b)]/2
Therefore, we have proved the cosine sum identity.
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