`text(Statements)`
`(i)` If `theta_1 , theta_2 , theta_3 , ........ , theta_n in R` and `i - sqrt(-1),` then
`(cos theta_1+ i sin theta_1)(cos theta_2+ i sin theta_2)(cos theta_3+ i sin theta_3)............(cos theta_n+ i sin theta_n)`
`= cso (theta_1+ theta_2+theta_3+.....+ theta_n) + isin(theta_1+theta_2+theta_3+.......+theta_n)`
`(ii)` If `theta in R , n in I` (set of integers) and `i=sqrt(-1),` then
`(cos theta + i sin theta)^n = cosn theta+ isin ntheta`
`(iii)` If `theta in R , n in Q` (set of rational numbers) and `i = sqrt(-1),` then
`cos n theta + i sin theta` is one of the values of `(cos theta + isin theta)^n.`
`text(Proof)`
`(I)` By Euler's formula `E^(i theta) = cos theta + i sin theta`
`LHS = ( cos theta)_1 + i sin theta_1 ) ( cos theta_2 + i sin theta_2) ( cos theta)_3 + i sin theta_3 ).......... ( cos theta_n + i sin theta_n)`
`= e^(i theta_1) . e^(i theta_2) . e^(i theta_3).............e^(i theta_n) = e^(i(theta_1+theta_2+theta_3+......+theta_n))`
`cos (theta_1+theta_2+theta_3+..........+theta_n)+ isin(theta_1+theta_2+theta_3+.........+theta_n) = RHS`
`(ii) theta_1 = theta_2 = theta_3=.....=theta_n = theta,` then from the above result (i),
`cos(theta+theta+theta+......+ text(up to n times))+ isin(theta+theta+theta+........text(up to n times))`
`i.e., (cos theta+ i sin theta)^n = cos n theta + isin ntheta`
`(iii)` Let `n = p/q,` where `p , q in I` and `q ne 0,` from above result (ii) , we have
`(cos(p/q theta) + i sin(p/q theta))^q = cos(((p/q theta)q)+isin((p/q theta))q)`
`= cos p theta + i sin p theta`
`=> cos (p theta)/q + i sin(ptheta)/q` is one of the values of `(cos p theta + isin p theta)^(1/q)`
`=> cos (p theta)/q + i sin (ptheta)/q` is one of the values of `[(cos theta + isin theta)^p]^(1/q)`
`=> cos (p theta)/q + i sin (ptheta)/q` is one of the values of `(cos theta + isin theta)^(p/q)`
`text(Important Result)`
`1. (cos theta - i sin theta)^n = cos n theta - i sin theta , AA n in I`
`2. (sin theta - i cos theta)^n =(i)^n (cos n theta - i sin theta) , AA n in I`
`3. (sin theta - i cos theta)^n =(-i)^n (cos n theta + i sin theta) , AA n in I`
`4. (cos theta + i sin phi)^n ne cos n theta + i sin phi , AA n in I`
[here, `theta ne phi` De-Moivre's theorem is not applicable]
`5. 1/(cos theta+ i sin theta) = (cos theta + i sin theta)^(-1)`
`= cos(- theta) + i sin(-theta) = cos theta - i sin theta`
`text(Statements)`
`(i)` If `theta_1 , theta_2 , theta_3 , ........ , theta_n in R` and `i - sqrt(-1),` then
`(cos theta_1+ i sin theta_1)(cos theta_2+ i sin theta_2)(cos theta_3+ i sin theta_3)............(cos theta_n+ i sin theta_n)`
`= cso (theta_1+ theta_2+theta_3+.....+ theta_n) + isin(theta_1+theta_2+theta_3+.......+theta_n)`
`(ii)` If `theta in R , n in I` (set of integers) and `i=sqrt(-1),` then
`(cos theta + i sin theta)^n = cosn theta+ isin ntheta`
`(iii)` If `theta in R , n in Q` (set of rational numbers) and `i = sqrt(-1),` then
`cos n theta + i sin theta` is one of the values of `(cos theta + isin theta)^n.`
`text(Proof)`
`(I)` By Euler's formula `E^(i theta) = cos theta + i sin theta`
`LHS = ( cos theta)_1 + i sin theta_1 ) ( cos theta_2 + i sin theta_2) ( cos theta)_3 + i sin theta_3 ).......... ( cos theta_n + i sin theta_n)`
`= e^(i theta_1) . e^(i theta_2) . e^(i theta_3).............e^(i theta_n) = e^(i(theta_1+theta_2+theta_3+......+theta_n))`
`cos (theta_1+theta_2+theta_3+..........+theta_n)+ isin(theta_1+theta_2+theta_3+.........+theta_n) = RHS`
`(ii) theta_1 = theta_2 = theta_3=.....=theta_n = theta,` then from the above result (i),
`cos(theta+theta+theta+......+ text(up to n times))+ isin(theta+theta+theta+........text(up to n times))`
`i.e., (cos theta+ i sin theta)^n = cos n theta + isin ntheta`
`(iii)` Let `n = p/q,` where `p , q in I` and `q ne 0,` from above result (ii) , we have
`(cos(p/q theta) + i sin(p/q theta))^q = cos(((p/q theta)q)+isin((p/q theta))q)`
`= cos p theta + i sin p theta`
`=> cos (p theta)/q + i sin(ptheta)/q` is one of the values of `(cos p theta + isin p theta)^(1/q)`
`=> cos (p theta)/q + i sin (ptheta)/q` is one of the values of `[(cos theta + isin theta)^p]^(1/q)`
`=> cos (p theta)/q + i sin (ptheta)/q` is one of the values of `(cos theta + isin theta)^(p/q)`
`text(Important Result)`
`1. (cos theta - i sin theta)^n = cos n theta - i sin theta , AA n in I`
`2. (sin theta - i cos theta)^n =(i)^n (cos n theta - i sin theta) , AA n in I`
`3. (sin theta - i cos theta)^n =(-i)^n (cos n theta + i sin theta) , AA n in I`
`4. (cos theta + i sin phi)^n ne cos n theta + i sin phi , AA n in I`
[here, `theta ne phi` De-Moivre's theorem is not applicable]
`5. 1/(cos theta+ i sin theta) = (cos theta + i sin theta)^(-1)`
`= cos(- theta) + i sin(-theta) = cos theta - i sin theta`