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Given condition
π/2 <A<π
so A liye in 2nd quadrant
in second quadrant sin@ and cosec@ is positive
and
tan@,cos@,sec@ ,cot@ is negative.
For B
3π/2 <B<2π
so B liye in 4th quadrant
in 4th quadrant
cos@ and sec@ is positive and rest
sin@,cosec@,tan@,cot@ is negative
Now by keeping Above explanation
Given
2sinA = 1
sinA = 1/2
A = Sin^-1sin(π/2 + π/6 )
A = 5π/12
Now
sqrt2 *cosB = 1
cosB = 1/sqrt2
B = cos^-1cos(3π/2 + π/4)
B = 7π/4
Now
(tanA + tanB )/(cosA + cosB)
(tan5π/12 + tan7π/4 )/(cos5π/12 + cos7π/12)
= (- 1/sqrt 3 - 1)/(- sqrt3/2 + 1/sqrt2)
= 2( 1+ sqrt3)/(2 - sqrt3)sqrt3
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