Prove that `sin(theta - pi/6) + cos(theta - pi/3) = sqrt3 sintheta`

Prove that tan(theta+(pi)/(6))+cot(theta-(pi)/(6))=(1)/(sin2theta-sin.(pi)/(3))

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

Prove that (sin(3pi-theta)cos(theta- pi/2)tan((3pi)/(2)-theta))/(cosec((13pi)/(2)+theta)sec(3pi+theta)cot(theta- pi/2))=cos^(4)theta

Prove the following:if 2sin(theta+pi/3)=cos(theta-pi/6) ,then tantheta+sqrt3=0

The value of the determinant |[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)|

The value of the determinant |(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))|

If 2 sin(theta+(pi)/(3))=cos(theta-(pi)/(6)) , then tantheta =

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

Prove that sin ^(2) theta + sin ^(2) (theta+(pi)/(3))+ sin ^(2) (theta-(pi)/(3))=(3)/(2)

Prove that cos theta + cos[(2pi)/3 + theta] + cos [(4pi)/(3) + theta] = 0 .