Prove that for all values of `theta`,
`|{:(sintheta,costheta,sin2theta),(sin(theta+(2pi)/(3)),cos(theta+(2pi)/(3)),sin(2theta+(4pi)/(3))),(sin(theta-(2pi)/(3)),cos(theta-(2pi)/(3)),sin(2theta-(4pi)/(3))):}|=0`

By using properties of determinats. Prove that- |(sintheta,costheta,sin2theta),(sin(theta+(2pi)/3), cos(theta + (2pi)/3) ,sin(2theta + (4pi)/3)),(sin(theta-(2pi)/3) ,cos(theta - (2pi)/3), sin (2theta - (4pi)/3))|= 0

sin theta + sin 5 theta = sin 3theta (0 le theta le 2pi)

. Let f(theta)=|{:(cos^2theta,costhetasintheta,-sintheta),(sinthetacostheta,sin^2theta,costheta),(sintheta,-costheta,0):}|,"find" f(pi/3) .

Prove that 4 sin theta sin(pi/3 + theta) sin (pi/3-theta) = sin 3 theta

cos3theta + cos2theta = sin ((3theta)/2)+ sin (theta/2), 0 le theta le 2pi

Prove that (sin theta+ sin 2theta + sin 3 theta)/(costheta+cos2theta+cos3theta)=tan 2 theta

Solve: 4[ sin^(4) theta+ sin^(4)(theta+(pi)/(4)) + sin^(4) (theta-(pi)/(4))]=5

Show : (2theta-sin2theta)/(theta^(2))(thetagt0) is maximum at theta=(pi)/(2)

Prove that, cos^(3) theta cos 3 theta + sin^(3) theta sin 3 theta = cos^(3) 2 theta

Prove that, (sin 3 theta)/(sin theta)-(cos 3 theta)/(cos theta)=2