Question 1133311
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I will solve it in two steps.



<U>Step 1</U>


<pre>
For brevity, let's denote x = sin(A)*cos(A).


{{{(sin(A) + cos(A))/(sin(A)*cos(A))}}} = {{{1/3}}}  ====>  square both sides  ====>  


{{{(sin^2(A) + 2*sin(A)*cos(A) + cos^2(A))/(sin(a)*cos(A))^2}}} = {{{(1 + 2x)/x^2}}} = {{{1/9}}}  ====>


9*(1 + 2x) = {{{x^2}}}

{{{x^2 - 18x - 9}}} = 0

{{{x[1,2]}}} = {{{(18 +- sqrt(18^2 + 4*9))/2}}} = {{{(18 +- sqrt(360))/2}}}.


Since the modulus of x,  |x|, must be less than 1, only the root   x = {{{(18 - sqrt(360))/2}}}  is the solution.


It implies  2x = 2*sin(A)*cos(A) = sin(2A) = {{{18 - sqrt(360)}}} = {{{18 - 6*sqrt(10)}}}.      (1)
</pre>


<U>Step 2</U>


<pre>
{{{sin^4(A) - cos^4(A)}}} = {{{(sin^2(A) - cos^2(A))}}}.{{{(sin^2(A) + cos^2(A))}}}  ====>  replace  {{{(sin^2(A) + cos^2(A))}}}  by 1  ====>


{{{sin^4(A) - cos^4(A)}}} = {{{sin^2(A) - cos^2(A)}}} = {{{-cos(2A)}}} = {{{-sqrt(1 - sin^2(2A))}}}.


Substitute here {{{sin^2(2A)}}}  from (1)  and continue


{{{sin^4(A) - cos^4(A)}}} = {{{-sqrt(1 - sin^2(2A))}}} = {{{-sqrt(1 - (18 - sqrt(360))^2)}}} = {{{-sqrt(1 - 324 +2*18*sqrt(360) - 360)}}} = {{{-sqrt(2*18*6*sqrt(10) - 683)}}} = 


= {{{-sqrt(216*sqrt(10)-683)}}} = -0.228 (approximately, with 3 correct decimal places).


<U>ANSWER</U>.  If  {{{(sin(A) + cos(A))/(sin(A)*cos(A))}}} = {{{1/3}}},  then  {{{sin^4(A) - cos^4(A)}}} = {{{-sqrt(216*sqrt(10)-683)}}} = -0.228 (approximately, with 3 correct decimal places).
</pre>

Solved.