Question 1126750


 What is the value of {{{cos(2alpha)}}} , if 


{{{sin(3alpha) =2 sin(alpha)}}}? 


use the following identity: 

{{{sin(3alpha) = 3 sin(alpha) cos^2(alpha) - sin^3(alpha)}}}

factor it: 

{{{sin(3alpha) =sin(alpha)( 3 cos^2(alpha) - sin^2(alpha))}}}

use identity  {{{sin^2(alpha)=1-cos^2(alpha)}}} 


{{{sin(3alpha) =sin(alpha)( 3 cos^2(alpha) - (1-cos^2(alpha)) )}}}

{{{sin(3alpha) =sin(alpha)( 3 cos^2(alpha) - 1+cos^2(alpha)) )}}}

{{{sin(3alpha) =sin(alpha)( 4cos^2(alpha) - 1 )}}}

then we have

{{{sin(alpha)( 4cos^2(alpha) - 1 )=2sin(alpha)}}}

{{{cross(sin(alpha))( 4cos^2(alpha) - 1 )=2cross(sin(alpha))}}}

{{{4cos^2(alpha) - 1 )=2}}}...simplify, divide by{{{ 2}}}

{{{2cos(2 alpha) - 1=1}}}

{{{2cos(2alpha) =2}}}

{{{ cos(2alpha) =1}}}