Solve for x the equation tan2x+1=1/cos2x if x is in interval (0,pi).

First, we'll re-write the tangent function as the ratio
sin 2x/cos 2x.

We'll re-write the
equation:

sin 2x/cos 2x + 1 = 1/cos
2x

We'll multiply by cos 2x all
over:

sin 2x + cos 2x =
1

We'll recall the double angle identities for sin 2x and
cos 2x:

sin 2x = 2 sinx*cos
x

cos 2x = ` cos^2 x - sin^2
x`

We'll recall the Pythagorean
identity:

`sin^2x + cos^2x =
1`

We'll re-write the
equation:

`2sin x*cos x + cos^2 x - sin^2 x = sin^2 x + cos
^2 x`

We'll reduce like
terms:

`2sin x*cos x - 2sin^2 x =
0`

We'll divide by 2 and we'll factor sin
x:

sin x(cos x - sin x) =
0

We'll cancel each
factor:

sin x = 0

x = 0, but
the value doesn't belong to the interval (0,`pi` )

cos x -
sin x = 0

We'll divide by sin x both
sides:

cot x - 1 = 0

cot x =
1

The cotangent function has positive values within the
interval (0;`pi`), only in the 1st quadrant.

x =
`pi/4`

Therefore,
the only solution of the equation, over the interval `(0,pi)` is
{`pi/4`}.