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Why is the secant function is an even function and the tangent and cosecant are odd functions?
I find it convenient to express other trigonometric functions in terms of sine and cosine - that tends to simplify things. The secant function is even because it is the reciprocal of the cosine function, which is even. The tangent function is the sine divided by the cosine - an odd function divided by an even function. Therefore it is odd. The cosecant is the reciprocal of an odd function, so it is naturally also an odd function.
Why cos negative results positive?
It doesn't really. Depending on the exact value of the argument, the cosine function can give both positive and negative results, for a negative argument. As to "why" the sine, or cosine, functions have certain values, just look at the function definition. Take points on a unit circle. The sine represents the y-coordinate for any point on the circle, while the cosine represents the x-coordinate for such a point. (There are also other ways to define the sine and the cosine functions.)
What is the derivative of negative cosine?
The derivative of negative cosine is positive sine.
What is cosine squared theta equal to'?
Cosine squared theta = 1 + Sine squared theta
How are ratios for sin and cosine alike?
sin(x) = cos(pi/2 - x). Thus sine is simply a horizontal translation of the cosine function. NB: angles are measured in radians.
