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

2 Answers
Jun 25, 2018

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

Jun 26, 2018

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