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

Thursday, October 29, 2015

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 = $\displaystyle{{\cos}}^{{2}}{x}-{{\sin}}^{{2}}$
x

We'll recall the Pythagorean
identity:

$\displaystyle{{\sin}}^{{2}}{x}+{{\cos}}^{{2}}{x}=$
1

We'll re-write the
equation:

$\displaystyle{2}{\sin{{x}}}\cdot{\cos{{x}}}+{{\cos}}^{{2}}{x}-{{\sin}}^{{2}}{x}={{\sin}}^{{2}}{x}+{\cos{}}$
^2 x

We'll reduce like
terms:

$\displaystyle{2}{\sin{{x}}}\cdot{\cos{{x}}}-{2}{{\sin}}^{{2}}{x}=$
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, $\displaystyle\pi$ )

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; $\displaystyle\pi$ ), only in the 1st quadrant.

x =
$\displaystyle\frac{\pi}{{4}}$

Therefore,
the only solution of the equation, over the interval $\displaystyle{\left({0},\pi\right)}$ is
{ $\displaystyle\frac{\pi}{{4}}$ }.

Notes