If `cot(theta-alpha),3cottheta,"cot"(theta+alpha)` are in A.P. and `theta` is not an integral multiple of `pi/2,` then the value of `(4sin^2theta)/(3sin^2alpha)=` _________

Answer

2

Explanation

Given `cot( theta -alpha ), 3 cot theta , cot ( theta + alpha)` are in A.P.
`rArr 6 cot theta = cot ( theta -alpha ) + cot ( theta + alpha)`
`rArr ( 6 cos theta ) /( sin theta ) = ( sin 2 theta)/( sin ( theta + alpha) sin ( theta - alpha ))`
`rArr 6 cos theta { sin ^(2) theta - sin ^(2) alpha}= 2 sin ^(2) theta cos theta `
`rArr 3 ( sin ^(2) theta - sin ^(2) alpha) = sin ^(2) theta `
`rArr 2 sin ^(2) theta = 3 sin^(2) alpha`
`therefore ( 2sin^(2) theta)/(3 sin^(2) alpha) =1`