Question 1072350
.
<pre>
sin(4x) = cos(2x)  --->  2sin(2x)*cos(2x) = cos(2x)  --->  2sin(2x)*cos(2x) - cos(2x)  = 0  --->  

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

The last equation deploys in two independent equations:


1.  cos(2x) = 0  ---->  x = {{{pi/4}}}, x = {{{3pi/4}}},  x = {{{5pi/4}}}  and  x = {{{7pi/4}}}



2.  2sin(2x)-1 = 0  ---->  sin(2x) = {{{1/2}}},  which implies 


    2x = {{{pi/6}}}  and/or   2x = {{{5pi/6}}}.


    This, in turn, implies that the original equation has 4 (four) solutions in the interval 0 <= x < {{{2pi}}}:


    x = {{{pi/12}}},  x = {{{5pi/12}}},  x = {{{13pi/12}}}  and  x = {{{17pi/12}}}.


<U>Answer</U>.  In all, there are 8 solutions:  x = {{{pi/4}}}, x = {{{3pi/4}}},  x = {{{5pi/4}}},  x = {{{7pi/4}}};    x = {{{pi/12}}},  x = {{{5pi/12}}},  x = {{{13pi/12}}},  x = {{{17pi/12}}}.
</pre>

Solved.



{{{graph( 330, 330, -1.5, 6.5, -1.5, 1.5,
          sin(4x), cos(2x)
)}}}


Plots y = sin(4x) (red) and y = cos(2x) (green)



To see more examples of solved trigonometry equations with detailed solutions, look into the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-simple-problems-on-trigonometric-equations.lesson>Solving simple problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-typical-problems-on-trigonometric-equations.lesson>Solving typical problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-more-complicated-problems-on-trigonometric-equations.lesson>Solving more complicated problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solved-problems-on-trigonometric-equations.lesson>Solving advanced problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/OVERVIEW-of-lessons-on-calculating-trig-functions-and-solving-trig-equations.lesson>OVERVIEW of lessons on calculating trig functions and solving trig equations</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Trigonometry: Solved problems</U>". 



The solution by "Boreal" is wrong, since it does not cover a lot of roots.