Question

Question #00111

Answer 1

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)`