if `cos^2theta= (m^2-1)/3 and tan^3\ theta/2 = tanalpha,` prove that `cos^(2//3) alpha+ sin^(2//3) alpha= (2/m)^(2//3)`

If acos^3theta+3acosthetasin^2theta=m and asin^3theta+3acos^2thetasintheta=n , then prove that: (m+n)^(2//3)+(m-n)^(2//3)=2a^(2//3)

If tan 2theta= sec 2 alpha , prove that sin 2theta=(1-tan^4 alpha)/(1+tan^4 alpha)

If 2 cos theta = sqrt3. prove that : 3 sin theta - 4 sin ^(3) theta =1

If tan beta = 3 tan alpha, prove that tan (alpha + beta) = (2sin 2beta)/(1+cos 2beta)

If cos ec theta - sin theta = m and sec theta - cos theta = n, then show that (m ^(2) n ) ^(2//3) + (mn ^(2)) ^( 2//3) = 1.

If msintheta=n sin(theta+2alpha), prove that tan(theta+alpha)cotalpha=(m+n)/(m-n)dot

If 3 tan theta tan phi=1 , then prove that 2 cos (theta+phi)=cos (theta-phi) .

If m tan(alpha-theta)/( cos^2 theta) = n (tan theta)/ (cos^2 (alpha - theta) the prove that 2 theta = alpha - [ tan^(-1) ((n-m)/(n+m)) tan alpha]

If sin theta=n"sin"(theta+2alpha) , prove that tan(theta+alpha)=(1+n)/(1-n)tanalpha .

If tan beta=cos theta tan alpha , then prove that tan^(2)""(theta)/(2)=(sin(alpha-beta))/(sin(alpha+beta)) .