09. Prove that cos(A+B)= cos Acos B-sin Asin B.
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Answer:
Let i,j be the unit vectors along OX and OY. OP and are drawn such that
∴∠XOP=A and ∠XOQ=(−B)=B
∴∠POQ=A+B
Take M,N on OP and OQ such that
∣
∣
∣
∣
OM
∣
∣
∣
∣
=
∣
∣
∣
∣
ON
∣
∣
∣
∣
=1 unit
Draw ML⊥OX
Now
OM
=
OL
+
LM
=cosAi+sinAj
Similarly
ON
=cos(−B)i+sin(−B)j=cosBi−sinBj
So
OM
.
ON
=(cosAi+sinAj).(cosBi−sinBj)=cosAcosB−sinAsinB ....(1)
but
OM
.
ON
=
∣
∣
∣
∣
OM
∣
∣
∣
∣
∣
∣
∣
∣
ON
∣
∣
∣
∣
cos(A+B)=cos(A+B) ...(2)
From (1) and (2)
cos(A+B)=cosAcosB−sinAsinB
Hence, proved.
solution
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