Ok so the questions were:

(a) How to differentiate tan(x) sin(x) and to sin(x)(tan²(x) + 2)

(b) How to differentiate cos(3x)/sin(x).

For (a), we use:

1. product rule d/dx(f(x) g(x)) = f'(x)g(x) + f(x) g'(x)
2. the known derivative for tan, which is d/dx(tan(x)) = sec²(x)
3. the known derivative for sin, which is d/dx(sin(x)) = cos(x)
4. the definition of the tangent function, namely tan(x) = sin(x)/cos(x)
5. the Pythagorean Theorem for tan and sec, namely sec²(x) = tan²(x) + 1

Putting these together:

d/dx(tan(x) sin(x))

= sec²(x) sin(x) + tan(x) cos(x) [Using (1), (2) and (3)]

= sec²(x) sin(x) + sin(x) [Using (4)]

= (sec²(x) + 1) sin(x) [By factoring out sin(x)]

= (tan²(x) + 2) sin(x) [Using (5)]

Have a go at (b) yourself, keeping in mind, firstly the quotient rule, and secondly the trigonometric identities given on the relevant Wikipedia page.