 Hello and welcome to the session. In this session I will discuss a question which says that determine one of the values of cos pi by 3 minus iota sine pi by 3 whole raised to power 13 by 2 whole upon cos pi by 3 plus iota sine pi by 3 whole raised to power 5 by 2. Now before starting the solution of this question we should know our result. And that is DeMauver's theorem. According to this if n is any integral positive or negative then cos theta plus iota sine theta whole raised to power n is equal to cos n theta plus iota sine n theta. Now if n is any fraction or negative then one of the values plus iota sine theta whole raised to power n is equal to cos n theta plus iota sine n theta. Now this result will work out as a key idea for solving out this question. And now we will start with the solution. And this we have to find one of the values of cos pi by 3 minus iota sine pi by 3 whole raised to power 13 by 2 pi by 3 by 2. Now using this result which is given in the key idea we will find one of the values which is 13 by 2. So the DeMauver's theorem pi by 3 minus iota sine pi by 3 whole raised to power 13 by 2 is equal to minus iota sine n theta. Now here n is 13 by 2 and theta is pi by 3. So it will be cos which is 13 by 2 theta which is pi by 3 which is 13 by 2 theta which is pi by 3. Now this is equal to cos 13 by by 6 minus iota sine 13 by by 6 minus 13 by by 6. Now here iota and s pi it becomes s minus 13 by by 6. Similarly we will find one of the values of cos by 3 whole raised is equal to so it will be pi by 2 theta which is pi by 3. So this is equal to in the given expression which is this the given expression will be equal to minus 13 by by 6. Now this beta is equal to sine beta it will be equal to sine beta into sine beta over 1 cos beta sine beta the whole into cos beta minus iota sine beta the whole. Now for this applying the formula of the whole which is h4 minus v square which is equal to 1 the whole into sine beta the whole which is equal to cos alpha plus iota the whole is beta plus iota sine minus beta equal to cos minus beta this beta is equal to minus 13 by by 6 minus beta which is equal to this 5 by by 6 which is further equal to minus 3 pi will be equal to minus 1 and will be equal to 0. Now putting these values here this will be equal to minus 1 minus iota into 0 which is further equal to now 0 so this will be minus 1. So the given expression is minus 1 so this is the solution of the given question and that's all for this session hope you all have enjoyed the session.