What is the value of cos2x if sinx=1/3 ?

The problem requires the double angle
identity:

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

We'll determine cos x, applying the Pythagorean
identity:

(cos x)^2 + (sin x)^2 =
1

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

(cos x)^2 = 1 - 1/9

(cos
x)^2 = (9-1)/9

(cos x)^2 =
8/9

We'll replace the value of (cos x)^2 into the double
angle identity:

cos 2x = 2*8/9 -
1

cos 2x = (16-9)/9

cos 2x =
7/9

The value of cos 2x, if sin x = 1/3, is:
cos 2x = 7/9.