Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2015  OctNov  (P19709/12)  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 

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 
or

Therefore;
For 
For 




For































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


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