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# 2tan10+tan50=2x ,tan20=tan50=2y,2tan10=tan70=2w and tan20=tan70=2z ,then which of the following is true   z>w>y>x w=x+y 2y=x z+x=w=y

somi teez
105 Points
4 years ago
for above ...sorry....something wrong went
okay now,
lets consider tan 20 – 2 tan 10=(sin 20*cos 10 -2 sin 10 * cos 20) / cos 10*cos 20
=((sin 30 +sin 10) –2 (sin 30 -sin 10 )) /2 cos 10*cos20
=4 sin ^3 10/(2 cos 10 *cos 20)   > 0   (as, domain of sin and cos >1 or
tan 20 > 2 tan 10 …....(1)
so, y>x and z>w
now, 2w -2y
=2 tan 10 +tan 70 – tan 20 – tan 50
=2 tan 10 + tan 50 = 2x which is positive
so, w=x+y  and w >y
plz approve it
somi teez
105 Points
4 years ago
few more things to add ,
from above ,w= x+y  , w>y  ,y>x  and z>w    it means z>w>y>x
also, 2y -z
tan 20 +tan 50 -1/2 tan 20 -1/2 tan 70
=1/2 tan 20 +tan 50 -1/2 tan 70
= 1/2(tan 20 +2 tan 50 – tan 70)
=1/2((sin 70 / (cos 20 * cos 50 ) ) – sin 20 / (cos 50 *cos 70 ))
=(sin 140 – sin 40) /(4 cos 20 cos 50 cos 70)   = 0
2y -z =0     so,2 y =z
approve it if you agree
thank you