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Question
Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.
Solution
Let `hat"OP"` and `hat"OQ"` be unit vectors making angles A and B, respectively, with positive direction of x-axis.
Then ∠QOP = A – B .....[From the figure]
We know `hat"OP" = vec"OM" + vec"MP"`
= `hat"i" cos "A" + hat"j" sin "A"` and `hat"OQ" = vec"ON" + vec"NQ" = hat"i" cos"B" + hat"j" cos"B"`.
By definition `hat"OP" * hat"QO" = |hat"OP"| |hat"OQ"| cos("A" - "B")`
= `cos("A" - "B")` ......(1) `(because hat"OP"|= 1 = |hat"OQ"|)`
In terms of components, we have
`hat"QP" * hat"OQ" = (hat"i" cos"A" + hat"j" sin "A")*(hat"i" cos"B" + hat"j" sin"B")`
= cosA cosB + sinA sinB .....(2)
From (1) and (2), we get
cos(A – B) = cosA cosB + sinA sinB.
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