Question 1083018
<pre>
There are infinitely many solutions.  I will assume that
you were asking how many solutions there are in the interval
{{{matrix(1,5,"[",0,",",2pi,")")}}}


{{{2sin^""(x) + 3cos^""(x)= 2}}}

{{{2sin^""(x) + 3sqrt(1-sin^2(x))=2}}}

{{{3sqrt(1-sin^2(x))= 2-2sin^""(x)}}}

{{{9(1-sin^2(x))=4-8sin^""(x)+4sin^2(x)}}}

{{{9-9sin^2(x)=4-8sin^""(x)+4sin^2(x)}}}

{{{-13sin^2(x)+8sin^""(x)+5=0}}}

{{{13sin^2(x)-8sin^""(x)-5=0}}}

{{{(13sin^""(x)^""+5^"")(sin^""(x)^""-1^"")=0}}}


{{{matrix(4,6,

13sin^""(x)^""+5^""=0,"","","","",sin^""(x)^""-1^""=0,
13sin^""(x)=-5,       "","","","",sin^""(x)^""=1,
sin^""(x)=-5/13,      "","","","",x=pi/2,
matrix(1,3,
x=cross(3.536383773),or,5.888394187),"","","","","")



}}}

There are 2 solutions in the interval {{{matrix(1,5,"[",0,",",2pi,")")}}}
for the one crossed out is extraneous and doesn't check.
The other two do.

Edwin</pre>