# prove that 1-cosA+cosB-cos(A+B)/1+cosA-cosB-cos(A+B)=tanA/2cotB/2

jitender lakhanpal
62 Points
12 years ago

hi veda

take the numerator of L.H.S

1-cosA+cosB-cos(A+B) simplifying 1-cosA = 2sin2  A/2 by half angle identity

cosB-cos(A+B) = 2sin(A/2)sin((A+2B)/2)  difference into product identity

2sin2  A/2 + 2sin(A/2)sin((A+2B)/2)  by simple trigonometric identities

by further simplifying

2sin(A/2)(sin A/2 + sin((A+2B)/2)

now applying sum to product identity

2sin(A/2)(2sin((A+B)/2)cosB/2-------------1

and then take denominator

1+cosA-cosB-cos(A+B)

similarly by half angle and sum into product identity we get

2cos2  A/2 + 2cos(A/2)cos((A+2B)/2)

by further simplifying

2cos(A/2)(cos A/2 - cos((A+2B)/2)

applying difference to product identity we get

2cos(A/2)(2sin((A+B)/2)sinB/2---------------2

now dividing 1 by 2

we get

 tanA/2cotB/2

so proved

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