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Question

if a = cos 2B + cos 2A.  , b= cos 2B - cos 2 A

c= sin 2A+ sin 2B,  d= sin = sin2A-sin2B

Then which of the first is true.

(A)  a/b= cot(A+B) cot(A-B)

(B) c/d= tan(A+B)/ tan(A-B)

(C ) b/ c= tan(A-B)

(D) none of these


Solution

At first we need to know about certain equation they are
sin A + sin B = 2 sin [ (A + B) / 2 ] cos [ (A - B) / 2 ] 

      sin A - sin B = 2 cos [ (A + B) / 2 ] sin [ (A - B) / 2 ]

    cos A + cos B = 2 cos [ (A + B) / 2 ] cos [ (A - B) / 2 ] 

    cos A - cos B = - 2 sin [ (A + B) / 2 ] sin [ (A - B) / 2 ]


So first look at the 1st option (a/b)
So
(a/b)=[cos2b+cos2a]/[cos2b-cos2a]
So applying the equation given above
(a/b)=[2cos(2a+2b)/2 cos(2a-2b)/2]/[2sin(2a+2b)/2 sin(2b-2a)/2]

=[cos(a+b)cos(b-a)]/[[sin(a+b)sin(b-a)]
So (a/b)=[cot(a+b)cot(a-b)]
So option a is correct


Now next option
c/d=[sin2a+sin2b]/[sin2a-sin2b]
​​​​​​
now applying the equation given above
c/d=[2sin(2a+2b)/2 cos(2a-2b)/2]/[2cos(2a+2b)/2 sin(2a-2b)/2]
Now simplifying and cancelling 2
c/d=[sin(a+b)cos(a-b)]/[cos(a+b)sin(a-b)]
c/d=tan(a+b)cot(a-b)
Ie.c/d=tan(a+b)/tan(a-b)

So option b is correct...


Now we look for 3rd option
b/c=[cos2b-cos2a]/[sin2a+sin2b]

Applying the formulas
b/c=[2sin(2a+2b)/2 sin(2b-2a)/2]/[2sin(2a+2b)/2 cos(2b-2a)/2]

Simplifying and cancelling those 2 we get c/d=(tan(b-a))
So our option c is not correct 


So option a and b are correct
 

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