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

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 10sin^4 alpha +15cos^4alpha=6 then the value of 9cosec^4 alpha + 8 sec^4 alpha-75 is

Show that 1 + cot^2 alpha / (1+ cosec alpha) = cosec alpha

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.

(cos alpha + 2 cos 3 alpha + cos 5 alpha )/(cos 3 alpha + 2 cos 5 alpha + cos 7 alpha ) = cos 3 alpha sec 5 alpha .

If sin 3 alpha =4 sin alpha sin (x+alpha ) sin(x-alpha ) , then

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).

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

The locus of the centre of the circle (xcos alpha + y sin alpha - a)^2 + (x sin alpha - y cos alpha - b)^2= k^2 if alpha varies, is

If sin alpha + sin beta = a and cos alpha + cos beta = b, prove that : cos (alpha-beta) = 1/2 (a^2 + b^2 -2)