Q. 38 B4.8( 6 Votes )

# If sin α + sin β = a and cos α + cos β = b, prove that

Given: sin α + sin β = a & cos α + cos β = b

Proof:

sin α + sin β = a

Squaring both sides, we get

(sin α + sin β)2 = a2

sin2 α + sin2 β + 2 sin α sin β = a2 ……(1)

cos α + cos β = b

Squaring both sides, we get

(cos α + cos β)2 = a2

cos2 α + cos2 β + 2 cos α cos β = b2 ………(2)

Adding equation 1 and 2, we get

sin2 α + sin2 β + 2 sin α sin β + cos2 α + cos2 β + 2 cos α cos β = a2 + b2

sin2 α + cos2 α + sin2 β + cos2 β + 2 sin α sin β + 2 cos α cos β = a2 + b2

1 + 1 + 2 sin α sin β + 2 cos α cos β = a2 + b2

{ sin2 x + cos2 x = 1}

2 + 2 sin α sin β + 2 cos α cos β = a2 + b2

2(sin α sin β + cos α cos β) = a2 + b2 – 2

We know,

sin A sin B + cos A cos B = cos (A – B)

Therefore,

Hence Proved

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