multiply the numerator and the denominator by 2
numerator will become 2               &
the denominator             =   2sin^2a+6sinacosa+10cos^2a
    taking a (sin^2a + cos^2a) aside  we get
                                  =   sin^2a +6sinacosa +9cos^2a +sin^2a + cos^2a
                                  =   (sina + 3 cosa)^2  + sin^2a +cos^2a
                                  =  (sina + 3 cosa)^2  + 1
now the expression becomes                      2/( (sina + 3 cosa)^2  + 1  )
for the maximum value of the expression  the denominator must be minimum 
and for denominator to be minimum , (sina + 3 cosa)^2  must be minimum 
the minimum value of , (sina + 3 cosa)^2  can be zero   when   tana = -3
therefore the maximum value of the expression        2/( (sina + 3 cosa)^2  + 1  ) =  2/(  0 +1 ) 
                                                                    = 2                   
 
if you want to aprreciate the solution , please click below on approve.
						

							
Dear Smit,
Sanatan is right!! Its the right answer.
						

							
cosec^2(a)+5(cos^2(a))