Question 1082320
<pre><b>
{{{ sin^""(x) + cos^""(x) = 1/2 }}}

Square both sides of the equation

{{{ (sin^""(x) + cos^""(x)^"")^2 = (1/2)^2 }}}

{{{ (sin^""(x) + cos^""(x)^"")(sin^""(x) + cos^""(x)^"") = 1/4 }}}

Use FOIL on the left

{{{sin^2(x) + sin^""(x)cos^""(x)^"" + cos^""(x)sin^""(x) + cos^2(x) = 1/4 }}}

The two middle terms are like terms, so they combine:

{{{sin^2(x) + 2sin^""(x)cos^""(x)^"" + cos^2(x) = 1/4 }}}

Swap the 2nd and 3rd term on the left

{{{sin^2(x) + cos^2(x) + 2sin^""(x)cos^""(x)^""  = 1/4 }}}

Recall the Pythagorean identity {{{sin^2(theta)+cos^2(theta)=1}}}

{{{1 + 2sin^""(x)cos^""(x)^""  = 1/4 }}}

Clear the fraction by multiplying through by 4

{{{4 + 8sin^""(x)cos^""(x)^""  = 1 }}}

Subtract 4 from both sides

{{{8sin^""(x)cos^""(x)^""  = -3 }}}

Divide both sides by 8

{{{sin^""(x)cos^""(x)^""  = -3/8 }}}

Edwin</pre></b>