Prove the identity: (cos α + cos β)2 + (sin α + sin β)2 = 4 co

1.	Prove the identity: (cos α + cos β)2 + (sin α + sin β)2 = 4 cos2(α – β)/2
Answer» Let us consider the LHS (cos α + cos β)² + (sin α + sin β)² Now, upon expansion, we get, (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 = 2 + 2 cos α cos β + 2 sin α sin β = 2 (1 + cos α cos β + sin α sin β) = 2 (1 + cos (α – β)) [since, cos (A – B) = cos A cos B + sin A sin B] = 2 (1 + 2 cos² (α – β)/2 – 1) [since, cos2x = 2cos² x – 1] = 2 (2 cos² (α – β)/2) = 4 cos² (α – β)/2 = RHS Thus proved.

Prove the identity: (cos α + cos β)2 + (sin α + sin β)2 = 4 cos2(α – β)/2

Answer»

Let us consider the LHS

(cos α + cos β)² + (sin α + sin β)²

Now, upon expansion, we get,

(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

= 2 + 2 cos α cos β + 2 sin α sin β

= 2 (1 + cos α cos β + sin α sin β)

= 2 (1 + cos (α – β)) [since, cos (A – B) = cos A cos B + sin A sin B]

= 2 (1 + 2 cos² (α – β)/2 – 1) [since, cos2x = 2cos² x – 1]

= 2 (2 cos² (α – β)/2)

= 4 cos² (α – β)/2

= RHS

Thus proved.