# 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;

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