Since we have to differentiate a fraction, we must use the quotient rule.

The quotient rule: If y = u/v, dy/dx = (vdu/dx - udv/dx)/v²

So we must work out each of the terms u, v, du/dx and dv/dx from the question:

u = cos(x), v = sin(x), du/dx = -sin(x) , dv/dx = cos(x)

Plugging these into the equation given by the quotient rule gives:

dy/dx = (sin(x)*-sin(x) - cox(x)*cos(x))/(sin(x))²

= -(sin(x)²+ cos(x)²)/(sin(x))²

= -1/(sin(x))²

Side note: this is equal to -(cosec(x))²

The answer can also be obtained by rewriting the question as y = cos(x)*(sin(x))^-1 and then using the product rule, with u = cos(x) and v = (sin(x))^-1 . This is always possible - so if time allows, the question can be repeated using each method (the quotient rule and the product rule). If the same answer is obtained from each method, then you know it's right!

Where do the kinematics equations (SUVAT) come from?

Related Maths A Level answers

integrate x^2(2x - 1)

How do you integrate ln(x)?

What are partial fractions for and how do I find them?

How do you differentiate y=cox(x)/sin(x)?