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Question

The number of solution(s) of the equation 16sin2x+16cos2x=10, where 0x2π is 


Solution

16sin2x+16cos2x=10
16sin2x+161sin2x=10
16sin2x+1616sin2x=10
Let 16sin2x=t
t+16t=10
t210t+16=0
(t2)(t8)=0
t=2 and t = 8

Now, 16sin2x=2 and 16sin2x=8 
24sin2x=21 and 24sin2x=23
4sin2x=1 and 4sin2x=3
sin2x=14 and sin2x=34
sinx=±12 and sinx=±32
x=π6,5π6,7π6,11π6
and x=π3,2π3,4π3,5π3   
Hence, number of solutions =8

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