tan2 x sin x sin x sec x csc x squareroot 2 sec x Solution

tan^2 x sin x = sin x sec x csc x = squareroot 2 sec x

a) tan^2xsinx = sinx

sinx(tan^2x -1)=0

sin x=0

In the interval [0, 2pi)

x= 0, pi

tan^2x -1 =0 ---> tanx = +/-1

tanx =1 ;x = pi/4 , 5pi/4

tanx = -1 ; x= 3pi/4 , 2pi-pi/4

x= 3pi/4 , 7pi/4

Solution : x = pi/4 , 5pi/4 , 3pi/4 , 7pi/4

d) secxcscx = sqrt2 secx

(1/cosx)(1/sinx) = sqrt2(1/cosx)

cosx/cosx(1/sinx) = sqrt2

1/sinx = sqrt2

sinx = 1/sqrt2

In the interval [0, 2pi)

x = pi/4 , pi -pi/4 = pi/4 , 3pi/4

Solution : x= pi/4 , 3pi/4