# If sin α + sin β =A And cos α + cos β =B, show that(i) (ii)

Given sin α + sin β =A And cos α + cos β =B.

A2 +B2 =(sin α + sin β)2 +(cos α + cos β)2

= sin2 α + sin2 β + 2 sin α sin β + cos2 α + cos2 β + 2 cos α cos β

= sin2 α + cos2 α + sin2 β + cos2 β + 2(sin α sin β + cos α cos β)

We know that cos(A -B) = cosA cosB + sinA sinB

A2 +B2 = 2 + 2 cos(α – β) …(1)

Then,

B2 –A2 =(cos α + cos β)2 –(sin α + sin β)2

= cos2 α + cos2 β + 2 cos α cos β –(sin2 α + sin2 β + 2 sin α sin β)

=(cos2 α – sin2 β) +(cos2 β – sin2 α) – 2cos(α + β)

= 2 cos(α + β) cos(α – β) + 2 cos(α + β)

= cos(α + β)(2 + 2 cos(α – β)) …(2)

From(1) And(2),

B2 –A2 = cos(α + β)(A2 +B2)

…(ii)

And

…(i)

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