Question

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

Answer 1

`x=pi/4+-pin`
`x=(3pi)/4+-pin`

Explanation:

I assume you mean solve it

`2csc^2x = 2sec^2x`

Set the expression equal to 0:
`2csc^2x- 2sec^2x=0`

Factor out the `2csc^2x` by dividing the left side of the equation by `2csc^2x`:

`2csc^2x(1- (2sec^2x)/(2csc^2x))=0`

Apply reciprocal identities:

`2csc^2x(1- (1/cos^2x)/(1/sin^2x))=0`

`2csc^2x(1- 1/cos^2x*sin^2x)=0`

`2csc^2x(1- sin^2x/cos^2x)=0`

Apply quotient identity:
`2csc^2x(1- tan^2x)=0`

Set factors equal to 0 and solve:
`tan^2x=1`
`tanx=+-1`
`x=pi/4+-pin`
`x=(3pi)/4+-pin`

`cscx=0`
No solution

Answer 2

`x = +- pi/4 + kpi`

Explanation:

`csc^2 x - sec^2 x = 0`
`(csc x - sec x)(csc x + sec x) = 0`
`(1/sin x - 1/cos x)(1/sin x + 1/cos x) = 0`
`(sin x - cos x)(sin x + cos x) = 0`
a. sin x - cos x = 0
Divide both sides by cos x
(condition `cos x != 0`, or `x != pi/2`, or `x != (3pi)/2`)
tan x = 1
Trig table and unit circle give -->
`x = pi/4 + kpi`
b. sin x + cos x = 0 --> tan x = -1
`x = -pi/4 + kpi`
General answer: `x = +- pi/4 + kpi`