How do you solve: (1+cot^2(x))(sin2x)=2 ?

1 Answer
Feb 22, 2018

#x=pik#

Explanation:

Identities.

#color(red)bb(cotx=cosx/sinx)#

#color(red)bb(sin2x=2sinxcosx)#

#color(red)bb(sin^2x+cos^2x=1)#

#(1+cos^2x/sin^2x)(2sinxcosx)=2#

Add fractions:

#(sin^2x+cos^2x)/sin^2x(2sinxcosx)=2#

#1/sin^2x(2sinxcosx)=2#

#(2sinxcosx)/sin^2x=2#

#(2sinxcosx)=2sin^2x#

#2sinxcosx-2sin^2x=0#

Factor:

#sinx(2cosx-2sinx)=0#

#sinx=0#

#x=arcsin(sinx)=arcsin(0)=>x=pik#

#2cosx-2sinx=0#

#(2sinx)/(2cosx)=0#

#tanx=0#

#x=arctan(tanx)=arctan(0)=>x=pik#

#x=pik#

#k in ZZ#