Tan x + 1/cos x = 2 cos x How do I solve this?

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Answer 1 · 2018-04-29T11:40:20

#x={(4k-1)pi/2,kinZZ}uu{kpi+(-1)^k*pi/6,kinZZ}#

Here,

#tan x + 1/cos x = 2 cos x#

#=>sinx/cosx+1/cosx=2cosx#

#=>sinx+1=2cos^2x#

#=>sinx+1=2(1-sin^2x)#

#=>sinx+1-2+2sin^2x=0#

#=>2sin^2x+sinx-1=0#

#=>2sin^2x+2sinx-sinx-1=0#

#=>2sinx(sinx+1)-1(sinx+1)=0#

#=>(sinx+1)(2sinx-1)=0#

#=>sinx+1=0or 2sinx-1=0#

#=>sinx=-1 or sinx=1/2#

#(i)sinx=-1=>color(blue)(x=(4k-1)pi/2,kinZZ#

#(ii)sinx=1/2=>sinx=sin(pi/6)#

#:.color(blue)(x=kpi+(-1)^k*pi/6,kinZZ#
Hence,
#x={(4k-1)pi/2,kinZZ}uu{kpi+(-1)^k*pi/6,kinZZ}#