 Hello and welcome to the session. In this session we will learn about DeLauro's theorem. Now DeLauro's theorem states that n is any integer positive or negative than theta plus iota sine theta for n is equal to plus iota sine and theta. Affection to values plus iota sine theta always to power n and theta plus it should be also noted that the modulus sine theta is 1 and the argument is less than equal to theta is less than equal to pi that is the value of theta varies from minus pi to pi. So these are the cases according to the DeLauro's theorem case 1 when n positive integer. So let us start with the case 1. Now by actual multiplication we have cos theta 1 plus iota sine theta 1 the whole into cos theta 2 plus iota sine theta 2 the whole is equal to theta 1 into cos theta 2 plus theta 1 sine theta 2 plus iota sine theta 1 cos theta 2 plus iota square sine theta 1 into sine theta 2 now iota square is minus 1 so it will become minus sine theta 1 into sine theta 2 now further this is equal to us theta 2 minus sine theta 1 sine theta 2 the whole iota into sine theta 2 plus theta 2 the whole cos b minus sine is sine b is equal to so it will be cos theta 1 plus theta 2 the whole theta 1 plus theta 2 the whole cos b is equal to sine a plus b. So by multiplying these two expressions we have achieved the result as cos theta 1 plus theta 2 the whole plus iota sine theta 1 plus theta 2 the whole theta 1 the whole into theta 2 the whole theta 3 the whole will be equal to cos theta 1 plus theta 2 plus theta 3 plus theta 1 plus theta 2 plus theta 3. Now proceeding this theta 1 the whole into theta 2 the whole is equal to theta 1 plus theta 2 plus theta 2 plus so on up to now putting the 1 is equal to theta 2 is equal to so on is equal to theta in the above relation over sine theta 1 is equal to cos theta 1 plus iota sine theta 1 and so on upto n factors will become r is equal to minus m where m is a positive integer. So in this case theta whole raise to power n will be equal to n written as 1 over i theta whole raise to power. Now using this result which we have proved in the case one as here n is a positive integer to 1 over. Now this is equal to we will visualize the denominator. So it will be m theta plus i the multiplication of these two expressions will become where m theta s minus 1. And we know that cos square theta plus sin square theta is equal to 1 therefore equal to minus m. So this is m is equal to minus m which implies theta is equal to cos cos now cos minus m theta will become cos m theta n theta will be equal to m theta. So it will become plus iota sin case one of case two. So from the case one and two we get that when n is an integer theta whole raise to power n is equal to cos n theta plus iota sin n theta. So we have discussed the cases when n is an integer positive this is the case three and here let m be a fraction positive or negative e over q where q is a positive integer and two positive or negative. Now from case one which we have discussed earlier this is an integer by q plus iota sin theta by q is equal to theta plus iota sin such we are having sin theta whole raise to power n is equal to cos n theta plus iota sin n theta plus iota sin theta by q the whole iota sin theta whole raise to power 1 by 2. Now iota sin theta is equal to now we have taken n is equal to p by q so it will become which is equal to iota sin theta whole raise to power the whole raise to power 1 by 2 to apply the case one and two that we have discussed earlier for the negative and positive integers and in both the cases we are getting the same result so here this value will become equal to 1 by 2 this is by the case one now in this result which we have obtained earlier we have got one of the values of cos theta plus iota sin theta whole raise to power 1 by q now here instead of theta we will write so now we know that theta plus iota theta 1 by q is n theta the whole that is replacing p by q by n p theta equal to cos theta plus iota sin theta whole raise to power n so cos n and n theta is one of the main theta is completely established for all rational values of n now let us discuss some corollary that is integral n theta is one of the values iota sin theta whole raise to power n that is whether n is integral fractional positive or negative cos n theta minus iota sin n theta is one of the values of cos theta minus iota sin theta whole raise to power n now let us discuss the corollary to theta is equal to plus minus iota sin theta by the nervous theorem minus 1 into theta which is cos minus theta plus minus iota sin n theta which is equal to this theta is cos theta minus theta is so this will become minus n theta let us if cos theta plus iota sin theta is equal to z sin theta is equal to 1 by z that means cos theta plus iota sin theta and cos theta now let us discuss the next corollary to theta can be written as by taking iota common it is iota into 1 by iota into sin theta which is further equal to iota into now 1 by iota is equal to minus iota sin theta plus cos theta the whole which is further equal to iota minus iota sin theta the whole equal to iota by z now sin theta stands for cos theta can be written as cos minus theta theta 1 into says theta 2 is equal to now says theta 1 can be written as theta 1 theta 1 the whole into says theta 2 theta 2 the whole which is further equal to cos theta 1 plus theta 2 the whole plus iota sin theta 1 plus theta 2 the whole which can be written as theta 1 plus theta 2 the whole theta 2 is equal to now says theta 1 is equal to cos theta 1 plus iota sin theta 1 whole upon so theta 2 is cos theta 2 plus iota sin theta 2 this multiplying the numerator and denominator by the conjugate of denominator this is equal to theta 2 minus iota sin theta 2 over cos theta 2 minus iota sin theta 2 square theta 2 plus sin square theta 2 will become 1 we get cos theta 1 plus iota sin theta 1 the whole theta 2 the whole 1 now this is equal to theta 2 theta 2 theta 2 is equal to cos theta 2 is equal to minus sin theta 2 is equal to minus theta 2 the whole plus iota sin theta 2 the whole which can be written as says theta 1 minus theta 2 the whole says theta 1 minus theta 2 is equal to says theta 1 minus theta to the whole and iota can be written as 2 has cos pi by 2 is 0 plus iota into i pi by 2 is 1, 0 plus iota which is equal to iota. Now further, so minus iota can be written as, so minus iota can be written as 6 minus pi by 2. Now the equation you have learnt about, the Dewey-Weirberg's theorem. And this completes our session. Hope you all have enjoyed the session.