# 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;  