# Solve 2csc^2x = 2sec^2x?

Jun 25, 2018

$x = \frac{\pi}{4} \pm \pi n$
$x = \frac{3 \pi}{4} \pm \pi n$

#### Explanation:

I assume you mean solve it

$2 {\csc}^{2} x = 2 {\sec}^{2} x$

Set the expression equal to 0:
$2 {\csc}^{2} x - 2 {\sec}^{2} x = 0$

Factor out the $2 {\csc}^{2} x$ by dividing the left side of the equation by $2 {\csc}^{2} x$:

$2 {\csc}^{2} x \left(1 - \frac{2 {\sec}^{2} x}{2 {\csc}^{2} x}\right) = 0$

Apply reciprocal identities:

$2 {\csc}^{2} x \left(1 - \frac{\frac{1}{\cos} ^ 2 x}{\frac{1}{\sin} ^ 2 x}\right) = 0$

$2 {\csc}^{2} x \left(1 - \frac{1}{\cos} ^ 2 x \cdot {\sin}^{2} x\right) = 0$

$2 {\csc}^{2} x \left(1 - {\sin}^{2} \frac{x}{\cos} ^ 2 x\right) = 0$

Apply quotient identity:
$2 {\csc}^{2} x \left(1 - {\tan}^{2} x\right) = 0$

Set factors equal to 0 and solve:
${\tan}^{2} x = 1$
$\tan x = \pm 1$
$x = \frac{\pi}{4} \pm \pi n$
$x = \frac{3 \pi}{4} \pm \pi n$

$\csc x = 0$
No solution

Jun 26, 2018

$x = \pm \frac{\pi}{4} + k \pi$

#### Explanation:

${\csc}^{2} x - {\sec}^{2} x = 0$
$\left(\csc x - \sec x\right) \left(\csc x + \sec x\right) = 0$
$\left(\frac{1}{\sin} x - \frac{1}{\cos} x\right) \left(\frac{1}{\sin} x + \frac{1}{\cos} x\right) = 0$
$\left(\sin x - \cos x\right) \left(\sin x + \cos x\right) = 0$
a. sin x - cos x = 0
Divide both sides by cos x
(condition $\cos x \ne 0$, or $x \ne \frac{\pi}{2}$, or $x \ne \frac{3 \pi}{2}$)
tan x = 1
Trig table and unit circle give -->
$x = \frac{\pi}{4} + k \pi$
b. sin x + cos x = 0 --> tan x = -1
$x = - \frac{\pi}{4} + k \pi$
General answer: $x = \pm \frac{\pi}{4} + k \pi$