If `y = sin^-1((sinalpha sin x)/(1-cos alpha sinx))` , find y'(0)

One side of a square makes an angle alpha with x axis and one vertex of the square is at origin. Prove that the equations of its diagonals are x(sin alpha+ cos alpha) =y (cosalpha-sinalpha) or x(cos alpha-sin alpha) + y (sin alpha + cos alpha) = a , where a is the length of the side of the square.

Find the value of f(alpha, beta) =|(cos (alpha + beta), -sin (alpha + beta), cos 2beta),(sin alpha, cos alpha, sin beta),(-cos alpha, sin alpha, cos beta)|

Let lambda and alpha be real. Then the numbers of intergral values lambda for which the system of linear equations lambdax +(sin alpha) y+ (cos alpha) z=0 x + (cos alpha) y+ (sin alpha) z=0 -x+(sin alpha) y -(cos alpha) z=0 has non-trivial solutions is

Statement I The system of linear equations x+(sin alpha )y+(cos alpha )z=0 x+(cos alpha ) y+(sin alpha )z=0 -x+(sin alpha )y-(cos alpha )z=0 has a not trivial solution for only one value of alpha lying between 0 and pi . Statement II |{:(sin x, cos x , cos x),( cos x , sin x , cos x) , (cos x , cos x , sin x ):}|=0 has no solution in the interval -pi//4 lt x lt pi//4 .

Locus of the point of intersection of lines x cosalpha + y sin alpha = a and x sin alpha - y cos alpha =b (alpha in R ) is

If A((sin alpha)/3 - 1,(cos alpha)/2 - 1) and B(1,1) alpha in [ -pi,pi] are two points on the same side of the line 3x - 2y +1 = 0 then alpha belongs to the interval

If (sin x)^y = (cos y)^x , find dy/dx .

Evaluate int_(0)^(pi)(x dx)/(1+cos alpha sin x) ,where 0lt alpha lt pi .