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If A+B=π/3 and cosA+cosB=1 then which of the following is true? cos(A-B)=1/√3 |cosA-cosB|=√(2/3) cos(A-B)=-1/3 |cosA-cosB|=1/2√3

If A+B=π/3 and cosA+cosB=1 then which of the following is true?
  1. cos(A-B)=1/√3  
  2. |cosA-cosB|=√(2/3)
  3. cos(A-B)=-1/3
  4. |cosA-cosB|=1/2√3

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3 Answers

Sunil Raikwar
askIITians Faculty 45 Points
9 years ago
Hello student, please check the solution of your question. Given below:

cosA+cosB=1
2cos(A+B/2)cos(A-B/2)=1
2(root3/2)cos(A-B/2)=1
cos(A-B/2)=1/root3
cos(A-B)=2cos2(A-B/2)-1= 2*1/3-1= -1/3

Thanks
Abhigyan
8 Points
9 years ago
Sir option 2 is also a answer. How?
Saurabh Kumawat
31 Points
5 years ago
cosA+cosB = 1
2cos[(A+B)/2]●cos[(A-B)/2] = 1
A+B = π/3
(A+B)/2 = π/6
2 cosπ/6●cos[(A-B)/2] = 1
Cos(A-B)/2 = 1/√3
Cos(A-B)=2cos^2[(A-B)/2] - 1
                = 2●1/3 - 1
                = -1/3
Now sin[(A-B)/2] = √{[1-cos(A-B)]/2}
                               = √(1+1/3)/2
                               =√4/6
                               =√(2/3)
Now |cosA - cosB|= | -2 sin[(A+B)/2]●sin[(A-B)/2] |
                                 = 2sinπ/6●sin[(A-B)/2]
                                 = 2●1/2●√(2/3)
                                 = √(2/3)
HENCE B&C option is correct

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