If `x sin alpha = y cos alpha,` prove that : `x/(sec 2alpha) + y/(cosec 2 alpha) = x`

A square lies above the X - axis and has one vertex at the origin . The side passing through the origin makes an angle alpha(o lt alpha lt pi//4) with the positive direction of the X - axis . Prove that the equation of its diagonals are , y (cos alpha - sin alpha ) = x (sin alpha + cos alpha ) , and y(sin alpha + cos alpha ) + x ( cos alpha - sin alpha ) = a where , is the length of each side of the square

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

If x sin ^ (3) alpha + y cos ^ (3) alpha = sin alpha cos alpha and x sin alpha-y cos alpha = 0 then x ^ (2) + y ^ (2) =

If P_(1) and p_(2) are the lenghts of the perpendiculars drawn from the origin to the two lines x sec alpha +y . Cosec alpha =2a and x. cos alpha +y. sin alpha =a. cos 2 alpha , show that P_(1)^(2)+P_(2)^(2) is constant for all values of alpha .

If p&q are lengths of perpendicular from the origin x sin alpha+y cos alpha=a sin alpha cos alpha and x cos alpha-y sin alpha=a cos2 alpha, then 4p^(2)+q^(2)

Eliminate alpha , if x = r cos alpha , y = r sin alpha .

If p and p' are the distances of the origin from the lines x "sec" alpha + y " cosec" alpha = k " and " x "cos" alpha-y " sin" alpha = k "cos" 2alpha, " then prove that 4p^(2) + p'^(2) = k^(2).

If x=r sin alpha cos beta, y= r sin alpha sin beta and z= r cos alpha , prove that x^(2)+y^(2) + z^(2) = r^(2).

If x=a sin alpha+b cos alpha and y=a cos alpha-b sin alpha then remove alpha