Trigonometry> If sinx+siny=root3(cosy-cosx) Prove that ...

If sinx+siny=root3(cosy-cosx)Prove that sin3x+sin3y=0.

Kevin Shah , 10 Years ago

Grade 12

2 Answers

Sunil Raikwar

Last Activity: 10 Years ago

Hello students, please check the solution of your question given below:
sinx+siny = root3 ( cosx-cosy)
2sin(x+y/2)cos(x-y/2) = root3 {2sin(x+y/2)sin(y-x/2)}
sin(x+y/2)[cos(x-y/2)-root3. sin(y-x/2)]=0
sin(x+y/2)= 0
x= -y.......................1
or cos(x-y/2)=root3. sin(y-x/2)
tan(x-y/2)=- 1/root3
then x-y/2 = - pie/6
x = - pie/3 +y...........2
putting the value of x from 1 & 2 in sin3x+sin3y we get the required answer.

Approved

Sumit Majumdar

Last Activity: 10 Years ago

Dear student,
According to the given relation, we would have:
$sinx+siny=\sqrt[3]{cosy-cosx}\Rightarrow sin^{3}x+sin^{3}y+3sin^{2}xsiny+3sinxsin^{2}y=cosy-cosx$ If we multiply both sides by a factor of 4, we get:
$4sin^{3}x+4sin^{3}y+12\left (1-cos^{2}x \right )siny+12sinx\left (1-cos^{2}y \right )=cosy-cosx\Rightarrow 3sinx-sin3x+3siny-sin3y+12siny+12sinx=12sinycos^{2}x+12sinxcos^{2}y+4cosy-4cosx$ This on simplifying gives us:
$sin3x+sin3y=15sinx+15siny+4cosx\left ( 1-3sinycosx \right )-4cosy\left ( 1+3sinxcosy \right )$ If the right hand side can be simplified to zero, we get the answer.
Regards
Sumit