# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | Oct-Nov | (P2-9709/02) | Q#8

Question

i.

a.   Prove the identity

b.   Hence prove that

ii.       By differentiating , show that if  then .

iii.       Using the results of parts (i) and (ii), find the exact value of

Solution

i.

a.

We are required to show that;

We know that;

provided that

b.

We are required to prove that;

We have demonstrated in i(a) that;

Therefore;

We have the trigonometric identity;

We have algebraic formula;

ii.

We are given that;

We are required to show that;

We know that;

provided that

Therefore;

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

Therefore;

If  and  are functions of , and if , then;

If , then;

Rule for differentiation of  is:

Rule for differentiation of  is;

We know that;

provided that

iii.

From i(b) we know that;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

We
have demonstrated in (ii) that;

Therefore reverse process of derivative, that is integral, of will yield .