Solve finding all solutions in 0 2 pi 15 tan x sec x square

Solve, finding all solutions in [0, 2 pi). 15) tan x sec x = squareroot 2

Solution

15) tanx secx =2

(sinx_/cosx)(1_/cosx)=2

(sinx_/cos²x)=2

sinx=cos²x2

sinx=(1-sin²x)2

sinx=2 -2sin²x

2sin²x +sinx=2

sin²x +(1_/2)sinx=1

sin²x +(1_/2)sinx +(1_/22)²=1 +(1_/22)²

(sinx +(1_/22))²=(9_/8)

(sinx +(1_/22))=(-3_/22),(sinx +(1_/22))=(3_/22)

(sinx )=(-4_/22),(sinx )=(2_/22)

sinx =(-2_/2),(sinx )=(1_/2)

sinx =-2,(sinx )=(1_/2)

sinx =-2 => no solution ,

(sinx )=(1_/2) => x =/4 ,3/4

therefore solution is  x =_/4 ,3_/4