Question 1038835
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Find all solutions of the equation in the interval [0, 2pi).

Sec^2x - secx = 2
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<pre>
{{{sec^2(x) - sec(x)}}} = {{{2}}}.


Introduce new variable u = seq(x). Then your equation becomes

{{{u^2 - u}}} = {{{2}}},

{{{u^2 - u - 2}}} = {{{0}}}.

Factor the left side. You will get

(u-2)*(u+1) = 0.

The solutions are 

u = 2  and/or  u = -1.

From this point you have two equations:

1)  sec(x) = 2  --->  {{{1/cos(x)}}} = 2  --->  cos(x) = {{{1/2}}}  --->  x = +/-{{{pi/3}}} + {{{2k*pi}}},  k = 0, +/-1, +/-2, . . . and

2)  sec(x) = -1  --->  {{{1/cos(x)}}} = -1  --->  cos(x) = {{{-1}}}  --->  x = {{{pi}}} + {{{2k*pi}}},  k = 0, +/-1, +/-2, . . .


<U>Answer</U>.  x = +/-{{{pi/3}}} + {{{2k*pi}}},  k = 0, +/-1, +/-2, . . . and
         x = {{{(2k+1)*pi}}},  k = 0, +/-1, +/-2, . . .
</pre>

Solved.