# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#4

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Question

i.       Prove the identity .

ii.       Hence solve the equation  for .

Solution

i.

We are given the identity;

We have the relation;

Therefore;

We have the trigonometric identity;

It can be rearranged as;

Hence;

We have the algebraic identity;

Therefore, we can write;

ii.

We are required to solve following equation.

From (i), we know that left hand side of above equation can be written as;

Using calculator;

We utilize the periodic property of   to find other solutions (roots) of :

 Symmetry Property

Hence;

Therefore, we have two solutions (roots) of the equation;

To find all the solutions (roots) over the interval , we utilize the periodic property of    for both these values of .

 Periodic Property or

Therefore;

 For For

For

Hence all the solutions (roots) of the equation  for  are;