Q. prove that the equation ,`(sin^-1)^3+(cos^-1)^3=alpha pi^3` has no root for `alpha<1/32`

Let alpha be a root of the equation (2sinx-cosx)(1+cosx)=sin^2x beta is a root of the equation 3cos^2x-10cosx+3=0 gamma be a root of the equation 1-sin2x=cosx-sinx 0<=alpha,beta,gamma<=pi/2 sin alpha+sin beta+sin gamma can be equal to (A) (14+3sqrt(2))/6 (B) 43226 (C) (3+4sqrt(2))/6 (D) (1+sqrt(2))/2

Eliminate alpha between the equations : ("sin"^(2)alpha)/(cosalpha)=a^(3) and (cos^(2)alpha)/("sin"alpha)=b^(3)

If alpha,beta are the roots of the equation,x^(2)-2x+3=0, find the equation whose roots are alpha^(3)-3 alpha^(2)+5 alpha-1 and beta^(3)-beta^(2)+beta+5

Q.Prove that from the equality (sin^(4)alpha)/(a)+(cos^(4)alpha)/(b)=(1)/(a+b) follows the relation; (sin^(8)alpha)/(a^(3))+(cos^(8)alpha)/(b^(3))=(1)/((a+b)^(3))

alpha is a root of equation ( 2 sin x - cos x ) (1+ cos x)=sin^2 x , beta is a root of the equation 3 cos ^2x - 10 cos x +3 =0 and gamma is a root of the equation 1-sin2 x = cos x- sin x : 0 le alpha , beta, gamma , le pi//2 sin alpha + sin beta + sin gamma can be equal to

The number of values of alpha in [-10pi, 10pi] for which the equations (sin alpha)x-(cos alpha)y+3z=0, (cos alpha)x+(sin alpha)y-2z=0 and 2x+3y+(cos alpha)z=0 have nontrivial solution is

alpha is a root of equation ( 2 sin x - cos x ) (1+ cos x)=sin^2 x , beta is a root of the equation 3 cos 2x - 10 cos x +3 =0 and gamma is a root of the equation 1-sin2 x = cos x- sin x : 0 le alpha , beta, gamma , le pi//2 cos alpha + cos beta + cos gamma can be equal to

alpha is a root of equation ( 2 sin x - cos x ) (1+ cos x)=sin^2 x , beta is a root of the equation 3 cos 2x - 10 cos x +3 =0 and gamma is a root of the equation 1-sin2 x = cos x- sin x : 0 le alpha , beta, gamma , le pi//2 sin (alpha - beta ) is equal to

Prove that the equation of a line passes through the point (a cos^(3)alpha , a sin^(3)alpha ) and perpendicular to the line x tan alpha+y=a sin alpha is x cos alpha-y sin alpha-a cos 2 alpha .