Question

Prove that Sin(A+B) = SinA.CosB+CosA.SinB​

Answer 1

Given :

• Sin(A+B) = SinA.CosB+CosA.SinB

To prove :

• Sin(A+B) = SinA.CosB+CosA.SinB

Solution :

$\sf \blue{\sin( A + B) \: = \dfrac{O}{H} = \dfrac{PR}{1}} \\ \\ \sf \pink{ \therefore \: sin (A + B = PR)} \\ \\ \sf \orange{Now \: sin \: A = \dfrac{O}{H} = \dfrac{PQ}{1} = PQ} \\ \\ \sf \green{ \dfrac{QS}{OQ} = QS = cos \: A \: sin \: B} \\ \\ \sf \red{cos \: A = \dfrac{R}{H} = \dfrac{OQ}{1} } \\ \\ \sf \blue{ \therefore \: cos \: A = OQ} \\ \\ \sf \pink{cos \: B = \dfrac{P}{H} = \frac{PT}{sin \: N} } \\ \\ \sf \orange{ \therefore \: PI = sin \: A \: cos \: B} \\ \\ \sf \green{Now \: PR \: = PT + QS \: \: \: \: \: \: \: as \: (TR \: = QS)} \\ \\ \implies\sf \red{PT + QS} \\ \\ \boxed{\sf {\blue{sin \: (A + B) = sin \: A \: cos \: B \: + cos \: A \: sin \: B \: \: \: \: \: \: \: \: \: \: \: \: Hence \: proved}}}$