SOLUTION: Simplify the given expression. sin(2x) cos(4x) − sin(4x) cos(2x)

Algebra ->  Trigonometry-basics -> SOLUTION: Simplify the given expression. sin(2x) cos(4x) − sin(4x) cos(2x)      Log On


   

Question 1204009: Simplify the given expression.
sin(2x) cos(4x) − sin(4x) cos(2x)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

+sin%282x%29+cos%284x%29+%E2%88%92+sin%284x%29+cos%282x%29
use identities
+sin%282x%29+=2cos%28x%29+sin%28x%29
+sin%284x%29=4cos%5E3%28x%29+sin%28x%29+%2B+4+%28-1%29+cos%28x%29+sin%5E3%28x%29
+cos%282x%29=cos%5E2%28x%29+-+sin%5E2%28x%29
+cos%284x%29+=cos%5E4%28x%29+-+6sin%5E2%28x%29+cos%5E2%28x%29+%2B+sin%5E4%28x%29

then we have
+sin%282x%29+cos%284x%29+%E2%88%92+sin%284x%29+cos%282x%29
=
=
=+-2+cos%28x%29+sin%5E5%28x%29+-+4cos%5E3%28x%29+sin%5E3%28x%29+-+2cos%5E5%28x%29+sin%28x%29
=
=+-2+sin%28x%29+cos%28x%29%281%29
=+-2+sin%28x%29+cos%28x%29

Reduced trigonometric form:
=+-sin%282+x%29

Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.

In this problem, they expect that you will use this standard Trigonometry identity

            sin(a)*cos(b) - cos(a)*sin(b) = sin(a+b),

which is valid for any/all angles  " a "  and  " b ".


Then the solution is in one line,  as it is shown below. 

    Let a = 2x, b= 4x.  Then sin(2x)*cos(4x) − sin(4x)*cos(2x) = sin(2x-4x) = sin(-2x) = -sin(2x).

Solved and completed.


=============================


The method shown in my post, is a traditional
and standard method solving similar problems.

If you will solve it as @MathLover1 does, your teacher will learn
that you do not know a standard way solving such problems.