# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | Q#8

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Question

i.
Express  in the form , where  and , giving the exact values of R and .

ii.       Hence show that

iii.       By differentiating , show that if  then .

iv.       Using the results of parts (ii) and (iii), show that

Solution

i.

We are given the expression;

We are required to write it in the form;

If  and are positive, then;

can be written in the form

can be written in the form

where,

and , , with

Considering the given equation, we have following case at hand;

can be written in the form

Comparing it with given equation Therefore

Therefore;

Finally, we can find , utilizing the equation;

Using calculator we can find that;

Therefore;

ii.

We are required to show that;

As demonstrated in (i), we can write;

Therefore, we can write the given equation as;

Since   provided that ;

Hence;

iii.

We are given that;

We are required to show that;

Since ;

Therefore;

If  and  are functions of , and if , then;

If , then;

Rule for differentiation of  is;

Rule for differentiation of  is;

We have the trigonometric identity;

Since   provided that ;

iv.

We are required to show that;

As demonstrated in (ii), we can write;

Therefore;

Rule for integration of  is:

Hence;