Question #00111

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Answer 1 · 2016-10-12T15:59:15

We have sine and cosine rule as follows

#sin(A+B)=sinAcosB+cosAsinB#

and

#cos(A+B)=cosAcosB-sinAsinB#

To prove sine rule using cosine rule.

Cosine rule being an identity it is valid for all real values of A and B.So we can put (#pi/2+A#) in place of A in the 2nd identity of cosine and get

#cos(pi/2+A+B)=cos(pi/2+A)cosB-sin(pi/2+A)sinB#

#=>-sin(A+B)=-sinAcosB-cosAsinB#

#=>sin(A+B)=sinAcosB+cosAsinB#

This is the sine rule.

Similarly we can also prove it for rule of #sin(A-B) and cos(A-B)#