given tan(x)=1/square root of 3 and sec(x)

Given that tanx =
1/sqrt(3)

We need to find sinx and
cosx

We know that tanx = sinx
/cosx

==> sinx/cosx =
1/sqrt(3)

Cross
multiply:

==> cosx = sqrt(3)*
sinx...........(1)

Also, we know that sin^2 x + cos^2 x =
1...........(2)

We will substitute with (1) into
(2):

==> sin^2 x + [sqrt3*sinx]^2 =
1

==> sin^2 x + 3sin^2 x =
1

==> 4sin^2 x =
1

Divide by4.

==> sin^2
x = 1/4

==> sinx = +-
1/2

==> cosx = sqrt3
sinx

==> cosx = +- sqrt3 /
2

But given that secx < 0 ==> cosx <
0

==> cosx = -sqrt3
/2

==> sinx =
-1/2