Question 1199966
.
Given: sin(x) - cos(x) = 1/(5^1/2) find the value of tan(x) + cot(x) + cos(2x)
~~~~~~~~~~~~~~~~~


<pre>
The given equality  sin(x) - cos(x) = {{{1/sqrt(5)}}}  implies after squaring


    sin^2(x) - 2sin(x)*cos(x) + cos^2(x) = {{{1/5}}}

    1 - sin(2x) = {{{1/5}}}

    1 - {{{1/5}}} = sin(2x)

    sin(2x) = {{{4/5}}}.      (1)


The value of  tan(x) + cot(x) + cos(2x) is

      {{{sin(x)/cos(x)}}} + {{{cos(x)/sin(x)}}} + cos(2x) = 

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

    = {{{2/sin(2x)}}} + cos(2x).


Substitute here sin(2x) = {{{4/5}}}  and  cos(2x) = {{{sqrt(1-sin^2(2x))}}} = {{{sqrt(1-(4/5)^2)}}} = {{{3/5}}},  based on (1).  You will get

    tan(x) + cot(x) + cos(2x) = {{{2/((4/5))}}} + {{{3/5}}} = {{{10/4}}} + {{{3/5}}} = {{{(5*10)/20}}} + {{{(4*3)/20}}} = 

                              = {{{(50+12)/20}}} = {{{62/20}}} = {{{31/10}}} = 3.1.    <U>ANSWER</U>


<U>ANSWER</U>.  If  sin(x) - cos(x) = {{{1/sqrt(5)}}}  then  tan(x) + cot(x) + cos(2x) = {{{31/10}}} = 3.1.
</pre>

Solved.