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

2 Answers
Hriman
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

Nghi N.
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#

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