# 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 .