if cos(a-b),cosa,cos(a+B) are in HP then prove that cos2a+1=cosb

If cos(a-b),cosa,cos(a+b) are in HP, then
1/cos(a-b), 1/cosa , 1/cos(a+b) are in A.P.

    1/cos(a+b) – 1/cosa = 1/cosa – 1/cos(a-b)
    [cosa – cos(a+b)] / cos(a+b).cosa = [cos(a-b) – cosa] / cos(a-b).cosa
    cos(a-b)[cosa – cos(a+b)] = cos(a+b)[cos(a-b) – cosa]
    cos(a-b).cosa – [cos(a-b).cos(a+b)] = [cos(a-b).cos(a+b)] – cos(a+b).cosa
    cos(a-b).cosa + cos(a+b).cosa = 2.cos(a-b).cos(a+b)
    cosa [cos(a-b) + cos(a+b)] = 2(cos2a – sin2b)
    cosa.(2.cosa.cosb) = 2(cos2a – sin2b)
    cos2a.cosb – cos2a = -sin2b
    cos2a.(cosb – 1) = -(1 – cos2b)
    cos2a.(cosb – 1) = (cosb – 1)(cosb + 1)
    cos2a – 1 = cosb

That means your question is wrong, it should have been like the above result. Hence proved.
Please check the signs.