# 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

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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; 