Question 1031139
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Please Find the general solution of:
sin(3x) - sin(x)= cos(2x)
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Apply the formula  {{{sin(alpha) - sin(beta)}}} = {{{2*sin((alpha-beta)/2)*cos((alpha+beta)/2)}}}  to the left side.

  (Regarding this formula, see the lesson <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions.lesson>Addition and subtraction of trigonometric functions</A> in this site).

You will get an equivalent equation

{{{2*sin(x)*cos(2x)}}} = {{{cos(2x)}}}.

Simplify it further:

{{{cos(2x)*(2sin(x)-1)}}} = {{{0}}}.

Then you have two equations:

(1)  cos(2x) = 0  --->  2x = {{{2k*pi +- pi/2}}}  --->  x = {{{k*pi +- pi/4}}}, k = 0, +/-1, +/-2, . . . ,   

and

(2)  2*sin(x) = 1  --->  sin(x) = {{{1/2}}}  --->  x = {{{pi/6 + 2k*pi}}}  and  x = {{{5pi/6 + 2k*pi}}}, k = 0, +/-1, +/-2, . . . .

The union of the sets (1) and (2) is your solution.
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