Given, S1={x∈[0,12π],sin5x+cos5x=1} S2={x∈[0,8π],sin7x+cos7x=1} (1) sin5x+cos5x=1 This satisfies when sinx=1 and cosx=0 ∴x=
π
2
,
5π
2
,
9π
2
,
13π
2
,
17π
2
,
21π
2
It also satisfies when sinx=0 and cosx=1 ∴x=0,2π,4π,6π,8π,10π,12π ∴ Accepted values of x in [0,12π] is =13 ∴n(S1)=13 (2) sin7x+cos7x=1 This satisfies when sinx=1 and cosx=0 For x∈[0,8π], possible values x=
π
2
,
5π
2
,
9π
2
,
13π
2
It also satisfies when sinx=0 and cosx=1 x∈[0,8π], possible values x=0,2π,4π,6π,8π ∴ Total accepted values of x in [0,8π] is =9 ∴n(S2)=9 ∴n(S1)−n(S2)=13−9=4