The number of solutions to the equations `cos^(4)x+(1)/(cos^(2)x)=sin^(4)x+(1)/(sin^(2)x)` in the interval `[0,2pi]` is

Options

A. 6
B. 4
C. 2
D. 0

Answer

B

Explanation

`cos^(4)x-sin^(4)x=(1)/(sin^(4)x)-(1)/(cos^(2)x)`
`(cos^(2)x-sin^(2)x)=((cos^(2)x-sin^(2)x))/(sin^(2)xcos^(2)x)`
`cos2x=(4cos2x)/(sin^(2)2x)`
`cos 2x=(1-4"cosec"^(2)2x)=0`
`cos 2x=0`
`2x=2npi+-(pi)/(2)`
`x=npi+-(4)/(pi)`
At `n=0,x=(pi)/(4)`
`n=1, n=(5pi)/(4),(3pi)/(4)`
`n=2 ,x=(7pi)/(4)`