Question

which of the following is incorrect??
1) 2cosx cosy = cos(x+y) + cos (x-y)
2) 2sinx siny = cos (x+y) - cos (x-y)
3) 2sinx cosy = sin (x+y) + sin (x-y)
4) 2cosx siny = sin (x+y) - sin (x-y)

Please please answer my Question correctly​

Answer 1

answer of this question is option number 2

Answer 2

We recall the formulae as follows:

$cos(x+y)=cosxcosy-sinxsiny\\cos(x-y)=cosxcosy+sinxsiny\\sin(x+y)=sinxcosy+cosxsiny\\sin(x-y)=sinxcosy-cosxsiny$

Given:

$2cosx cosy = cos(x+y) + cos (x-y)\\ 2sinx siny = cos (x+y) - cos (x-y)\\ 2sinx cosy = sin (x+y) + sin (x-y)\\2cosx siny = sin (x+y) - sin (x-y)$

We take the first equation,

RHS is as follows:

    $cos(x+y)+cos(x-y)=cosxcosy-sinxsiny+cosxcosy+sinxsiny$

                                          $=2cosxcosy$

Thus first equation is correct

Now we take second equation,

RHS is as follows:

     $cos(x+y)-cos(x-y)=cosxcosy-sinxsiny-cosxcosy-sinxsiny$

                                           $=-2sinxsiny$

This equation is incorrect

 We take third equation,

RHS is as follows:

       $sin(x+y)+sin(x-y)=sinxcosy+cosxsiny+sinxcosy-cosxsiny$

                                             $=2sinxcosy$

Third equation is also correct

Now we take fourth equation,

Consider its RHS,

       $sin(x+y)-sin(x-y)=sinxcosy+cosxsiny-sinxcosy+cosxsiny$

                                             $=2cosxsiny$

Thus, second equation is incorrect

                            $2sinxsiny=cos(x+y)-cos(x-y)$ is incorrect