# 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

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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 .      