It is a non-linear function of theta.
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If sine theta is 0.28, then theta is 16.26 degrees. Cosine 2 theta, then, is 0.8432
It is 1.
For such simplifications, it is usually convenient to convert any trigonometric function that is not sine or cosine, into sine or cosine. In this case, you have: sin theta / sec theta = sin theta / (1/cos theta) = sin theta cos theta.
Tangent (theta) is defined as sine (theta) divided by cosine (theta). In a right triangle, it is also defined as opposite (Y) divided by adjacent (X).
The properties of the cosine and sine function are based on the X and Y coordinates of a point on a circle that has a radius of 1 and a center at the origin (X=0,Y=0). If the angle of the line from the origin to the edge of the circle, at any arbitrary point (X,Y), with respect to the X axis is theta, then sine(theta) is X, and cosine(theta) is Y.