# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#7

Question

i.       By differentiating  , show that if y = cot x then

ii.       Hence, show that

By using appropriate trigonometrical identities, find the exact value of

iii.

iv.

Solution

i.

We are given;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

If  and  are functions of , and if , then;

Let  and , then;

Rule for differentiation of  is;

Rule for differentiation of  is;

We have the trigonometric identity;

provided that

Therefore, if;

provided that

Hence;

For;

As demonstrated above;

Hence;

ii.

As we have demonstrated in (i) that;

Therefore, the inverse of differentiate ie integral must be as;

Hence;

provided that

iii.

Utilizing the identity;

We can write;

Rule for integration of  is:

As demonstrated in (ii);

Hence;

Rule for integration of  is:

iv.

We have the trigonometric identity;

Rule for integration of  is:

provided that

We have shown in (ii) that;

Therefore;