Prove that sin (a + b) . cos (a - b) = sina cosa + sinb cosb
Answers
Step-by-step explanation:
How do you prove that sin(A+B) + cos(A-B) = (sinA+cosA) (sinB+cosB)?
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Answer to is already given, as anyone knows to prove identity you have to simplify any of the sides and without using any assumptions other side should be obtained. If you are very good in mathematics starting side is not important but for average ones better start from the side where you have more terms than other side. Most importantly you have to know all formulas related to the question otherwise definitely it may be difficult to minimize number of steps.
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Others have answered well.
But to answer How.
You would start working on one side (or the other, or both but separately)
It is still valid to start on both sides and work to the middle; many see it as “sloppy” but that isn’t true, it is resourceful. Using both sides to aid you in your quest. Because that is what a proof is; a quest.
Your task is to use known identities and basic arithmetic rules to make one side look like the other. If you are doing it right, each step must be justified with words to ensure the reader doesn’t get lost. (and you can’t refute the claim about ‘hand waving’ if someone
This is a simple exercise. Start with right hand side.
RHS= (sin A+cos A) (sin B+cos B)
= sin A. sin B+sin A. cosB+cos A. sin B+cos A. cos B
= [sin A. cosB+cos A. sin B] +[cos A. cos B+sin A. sin B]
= sin (A+B) + cos (A-B)
= LHS
QED