SOLUTION: Find {{{ sin2x }}} if {{{ secx=-4 }}} and {{{ sinx<0 }}}

Since the secant and the sine of x are both negative, that means
x is in the third quadrant. between 180° and 270° in degress,
or between  and  in radians.  So therefore
2x will be between 360° and 540° which is actually the 1st and
2nd quadrants.  The sine is positive in both the 1st and 2nd
quadrants, so the answer will be posotive.

We need these identities:

1. 
2.  
3.  
 
First we'll find  using 3.



Next we'll find  using 2.




 










However, since  we take the negative sign,



Now we can substitute in 1:









Edwin