how to solve cos^3x csc^3x tan^3x=csc^2x-cot^2x ?

2 Answers
Apr 3, 2018

x=0,pi,2pi,pi/4,5pi/4

Explanation:

cos3xtimes1/(sin3x)timestan3x=1/sin(2x)-cos(2x)/sin(2x)

1=(1-cos2x)/sin(2x)

sin(2x)=(1-cos(2x))

2sinxcosx=(cosx)^2+(sinx)^2-(cosx)^2+(sinx)^2

2sinxcosx=2(sinx)^2

(sinx)^2-sinxcosx=0

sinx(sinx-cosx)=0

sinx=0 or sinx-cosx=0

sinx=0 or tanx=1

x=0,pi,2pi,pi/4,5pi/4


  • I'm giving the answers between 0 and 2pi

Apr 3, 2018

Verified below

Explanation:

Using:
cscx=1/sinx
cotx= 1/tanx
1+cot^2x= csc^2x

Start:
cos^3x csc^3x tan^3x=csc^2x-cot^2x

cos^3x/sin^3x*tan^3x=csc^2x-cot^2x

cot^3 tan^3x=csc^2x-cot^2x

1/(cancel(tan^3x))* cancel(tan^3x)=csc^2x-cot^2x

1= csc^2x-cot^2x

1+csc^2x-csc^2x= csc^2x-cot^2x

1+csc^2x-(1+cot^2x)= csc^2x-cot^2x

cancel(1)+csc^2xcancel(-1)-cot^2x=csc^2x-cot^2x

Graph of csc^2x-cot^2x
graph{(cscx)^2-(cotx)^2 [-9.96, 10.04, -5.04, 4.96]} = csc^2x-cot^2x#

Graph of cos^3x csc^3x tan^3x
graph{(cosx)^3(cscx)^3(tanx)^3 [-9.96, 10.04, -5.04, 4.96]}