Find sinx2 cosx2 and tanx2 from the given information secx

Find sin(x/2), cos(x/2), and tan(x/2) from the given information sec(x) = 4/3, 270degree

sec(x) = 4/3 and x is QIV (4th quadrant so only cos and sec are positive)

cos(x) = 1/sec(x)

so cos(x) = 3/4

now by sin^2 (x) +cos^2 (x) =1 (identity)

sin^2 (x) + (3/4)^2 =1

sin^2(x) = 1- 9/16

sinx = -sqrt(7)/4 ( since x is in QIV)

now cos2x = cos^2(x) -sin^2(x)

cos2x = cos^2(x) -(1-cos^2(x))

cos2x = 2cos^2(x) -1

plug x=x/2

cosx = 2cos^2(x/2) -1

3/4 +1 = 2cos^(x/2)

7/4 = 2cos^2(x/2)

cos^2(x/2) = 7/8

cos(x/2) = sqrt(7/8)

now sin2x = 2sinx.cosx

plug x=x/2

sinx = 2sin(x/2).cos(x/2)

-sqrt(7)/4 = 2sin(x/2) .sqrt(7/8)

-1/4 = 2sin(x/2). 1/sqrt(8)

-1/8 = sin(x/2)/sqrt(8)

sin(x/2) = -1/sqrt(8)

tan(x/2) = sin(x/2)/cos(x/2)

= [-1/sqrt(8)] / [sqrt(7)/sqrt(8)]

= -1/sqrt(7)