Show that `((1+cosphi+isinphi)/(1+cosphi-isinphi))^n=cosphinphi+i sin nphi`

(1+sin phi+i cos phi)/((1+sin phi-i cos phi)^(n))=cos(n(pi)/(2)-n phi)+i sin(n(pi) /(2)-n phi)

Prove that ((1+cos theta+i sin theta)/(1+cos theta-i sin theta))^(n)=cos n theta+i sin n theta

Show that AB = BA in each of the following cases: (i) A=[{:(costheta,-sintheta),(sintheta,costheta):}]" and "B=[{:(cosphi,-sinphi),(sinphi,cosphi):}] (ii) A=[{:(1,2,1),(3,4,2),(1,3,2):}]" and "B=[{:(" "10,-4,-1),(-11," "5," "0),(" "9,-5," "1):}] (iii) A=[{:(1,3,-1),(2,2,-1),(3,0,-1):}]" and "B=[{:(-2,3,-1),(-1,2,-1),(-6,9,-4):}]

The determinant |(cos(theta+phi),-sin(theta+phi),cos2phi),(sintheta,costheta,sinphi),(-costheta,sintheta,cosphi)| is

If a^2+b^2+c^2=1, then |(a^2+(b^2+c^2)cosphi, ab(1-cosphi), ac(1-cosphi)), (ba(1-cosphi), b^2+(c^2+a^2)cosphi, bc(1-cosphi)), (ca(1-cosphi), cb(1-cosphi), c^2+(a^2+b^2)cosphi)| is independent of

If ((1 + cos theta + i sin theta ) /(i+ sin theta + i cos theta ))= cos ntheta + i sin n theta then n is equal to :

If quad f(x)=(cos x+i sin x)(cos2x+i sin2x)(cos3x+i sin3x)cos nx+i sin nx and f(1)=1, then f(1) is equal to (n(n+1))/(2) (b) {(n(n+1))/(2)}^(2)(c)-{(n(n+1))/(2)}^(2) (d) none of these