Given that Sin(x+y)=3/4 and Sin(x-y)=1/2 ,prove that Tanx=5Tany
Answers
Answer:
Sin (x+y) = 1
=> x+y = 90°…………..…..(1)
Sin (x-y) = 1/2
=> x-y = 30° …………..…..(2)
By adding eq(1) & (2)
=> 2x = 120°
=> x = 60°
=> y = 30°
So, tan( x+2y) = tan ( 60 +60) = tan 2*60 =
Since tan 2x = (2 tan x) / ( 1-tan²x)
=> tan 2*60° = ( 2*tan 60°) / ( 1- tan² 60°)
=> tan 120° = 2* √3 / ( 1- 3 )
=> tan 120° = 2√3 / -2
=> tan ( x+2y) = -√3 ●●●●●●●●(3)
Now, tan( 2x + y) = tan( 120 + 30 ) = tan 150°
= tan {180 + ( -30) }
Since tan ( a + b)
= ( tan a + tan b )/(1- tan a.tanb)
=> tan ( 180 +(-30) )
= {tan 180° + tan (-30°) } / ( 1- tan180° * tan(-30°)
= {0 + (sin -30° / cos -30° )} / (1–0)
=( 1/2 )/ (- √3/2)
= 1/ -√3
=> tan( 2x + y) = 1/ -√3 ●●●●●●●●●●●● (4)
So, by (3) & (4)
tan( x+2y) . tan( 2x + y) = -√3 *( 1/-√3)
= 1 ●●●●●●●● ANS
Option (A) is right
FIITJEE's home based online exam.
The question is already answered, I’ll just like to add that these questions are best solved by quickly checking some values satisfying the conditions.
For example from the given conditions -
sin(x+y) = 1 and sin (x-y) = 1/2 it is easy to see that x = π/3 and y = π/6 satisfy both the conditions.
Then solving tan(x+2y)tan(2x+y) can be easily solved by putting x = π/3 and y = π/6
Now, since sin(x−y)>0
x>y for principle value branchx−y=300
and x+y=900
Solving, we get ; x=600 and y=300.
NOTE : this is for the principal value branch (taken for convenience).
So, x+2y=1200
And y+2x=1500
ans : 1 (A)
Hope you understood .
Regards.
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