Find `theta`

`r^(n)=a^(n)cos theta`
`r^(n)=b^(n)sin theta`

(cos theta + i sin theta) ^ (n) = cos n theta + i sin n theta

If n is a positive integer,prove the following (1+cos theta+i sin theta)+(1+cos theta-i sin theta)^(n)=2^(n+1)cos^(n)((theta)/(2))cos((n theta)/(2))

Consider (1+sin theta+sin^(2)theta)^(n)=sum_(r=0)^(2n)a_(r) (sin theta)^( r ), theta in R . a_(n+1)+a_(n+2)+…+a_(2n-1) equals

If A=[cos theta-sin theta sin theta cos theta], then A^(T)+A=I_(2), if theta=n pi,n in Z(b)theta=(2n+1)(pi)/(2),quad n in Z(c)theta=2n pi+(pi)/(3),quad n in Z(d) none of these

Consider (1+sin theta+sin^(2)theta)^(n)=sum_(r=0)^(2n) a_(r)(sin theta)^( r ), theta in R . The value of a_(0)+2a_(1)+3a_(2)+…+(2n+1)a_(2n) is

Consider (1+sin theta+sin^(2)theta)^(n)=sum_(r=0)^(2n)a_(r)(sin theta)^( r ), theta in R . a_(0)^(2)-a_(1)^(2)+a_(2)^(2)-…a_(2n)^(2) is equal to

If P_(n)=cos^(n)theta+sin^(n)theta and Q_(n)=cos^(n)theta-sin^(n)theta, then which of the following is/are true.

If quad P_(n)=cos^(n)theta+sin^(n)theta and Q_(n)=cos^(n)theta-sin^(n)theta then which of the following is/are true.

If A=[cos theta i sin theta i sin theta cos theta], then prove by principle of mathematical induction that A^(n)=[cos n theta i sin n theta i sin n theta cos n theta] for all n in N.