Let `f(x)=sin^(23)x-cos^(22)xa n dg(x)=1+1/2tan^(-1)|x|` . Then the number of values of `x` in the interval `[-10pi,8pi]` satisfying the equation `f(x)=sgn(g(x))` is __________

Answer

9

Explanation

`g(x)=(1)/(2)tan^(-1)|x|+1 or sgn (g(x))=1`
` or sin^(23)x-cos^(22)x=1`
`or sin^(23)x=1+cos^(22)x`
which is possible if `sinx=1 and cosx =0.` Therefore,
`x=2 n pi+(pi)/(2),n in Z`
Hence, `-10pi le 2n pi+(pi)/(2) le 8pi or -(21)/(4) le n le (15)/(4)`
`or -5 le n le 3`
Hence, number of values of `x=9`