Question 1046263
what is the minimum value of sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)
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<pre>
Let me introduce c = cos(x) and s = sin(x) for brevity.

Then

{{{sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)}}} =


= {{{s^2 + c^2}}} + {{{1/c^2}}} + {{{1/s^2}}} + {{{s^2/c^2}}} + {{{c^2/s^2}}} =    ( replace  {{{s^2 + c^2}}}  by  1)

= {{{1 + (s^2 + c^2)/(s^2*c^2) +  (s^4 + c^4)/(s^2*c^2)}}} = {{{1}}} + {{{1/(s^2*c^2)}}} + {{{((s^4 + 2s^2*c^2 + c^4) - 2s^2c^2)/(s^2*c^2)}}} = 

= {{{1}}} + {{{1/(s^2*c^2)}}} + {{{((s^2 + c^2)^2 - 2s^2c^2)/(s^2*c^2)}}} = {{{1}}} + {{{1/(s^2*c^2)}}} + {{{(1 - 2s^2c^2)/(s^2*c^2)}}} = {{{1}}} + {{{2/(s^2*c^2)}}} - {{{2}}} = {{{2/(s^2*c^2) -1}}}.

Now,  {{{s^2*c^2}}} = {{{sin^2(x)*cos^2(x)}}} = {{{(1/4)*(2*sin(x)*cos(x))^2}}} = {{{(1/4)*sin^2(2x)}}} has the maximum {{{1/4}}}.


Therefore,  {{{sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)}}}  has the minimum equal to {{{2/((1/4)) - 1}}} = 8 - 1 = 7.

<U>Answer</U>.  {{{sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)}}}  has the minimum of 7.
</pre>

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -2.5, 8.5, -2.5, 20.5,
          (sin(x))^2+(cos(x))^2+(1/cos(x))^2+(1/sin(x))^2+(sin(x)/cos(x))^2+(cos(x)/sin(x))^2
)}}}


Plot y = {{{sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)}}}

  </TD>
  </TR>
</TABLE>