answer:Absolutely! The derivative of the trigonometric function sin(x) is actually quite straightforward. The derivative of sin(x) with respect to x is cos(x). So if f(x) = sin(x), then the derivative f'(x) = cos(x). This is a standard result, and is one of the basic derivatives in calculus that's usually memorized. Similarly, the derivative of cos(x) is -sin(x).

answer:Absolutely! The derivative of the trigonometric function tan(x) can be found using the quotient rule since tan(x) = sin(x)/cos(x), or it can be memorized as a standard derivative. The derivative of tan(x) with respect to x is sec^2(x). So if f(x) = tan(x), then the derivative f'(x) = sec^2(x). This is a standard result in calculus. The secant function, sec(x), is defined as 1/cos(x), and sec^2(x) means (sec(x))^2.

answer:Certainly, I'd be happy to help with that! The integral of the trigonometric function cos(x) with respect to x is sin(x). So if g(x) = cos(x), then the integral ∫g(x) dx = sin(x) + C. Remember to include the + C at the end. This is the constant of integration, which can be any constant, and is added because the derivative of a constant is zero. Like with derivatives, there are standard results for the integrals of basic trigonometric functions that are usually memorized. For example, the integral of sin(x) is -cos(x) + C.

answer:Sure, I'd be happy to help with that! The integral of the secant function, sec(x), is a standard result in calculus and it is a bit more complex compared to the basic trigonometric functions. The integral of sec(x) is ln|sec(x) + tan(x)| + C. So, if h(x) = sec(x), then the integral ∫h(x) dx = ln|sec(x) + tan(x)| + C. Here, ln denotes the natural logarithm, |...| denotes absolute value, and C is the constant of integration. This result is not as straightforward as some of the other trigonometric functions, and it's typically one that students look up or use a software to compute until they become familiar with it. The proof of this result involves some clever manipulations and substitutions.