51. If a cos 0 –b sin 0 = x and a sin 0 + b cos 0 = y, then prove that a + b- = P.

If x = a cos θ + b sin θ and y = a sin θ − b cos θ , then prove that y^2(d^2y)/(dx^2)-x(dy)/(dx) +y=0

If y = A sin x + B cos x , the prove that (d^2 y)/(dx^2) + y = 0

If sin x + sin y + sin z = 0 and cos x + cos y + cos z = 0 then prove that cot (x+y)/2 = cot z and cos (x-y)/2 = pm 1/2

If y = a cos m x + b sin m x , prove that y _2 +m^2 y = 0

If y= A sin x+ B cos x , then prove that (d^2y)/(dx^2)+y=0 .

If x=a cos theta+b sin theta and y=a sin theta-b cos theta, then prove that y^(2)(d^(2)y)/(dx^(2))-x(dy)/(dx)+y=0

If y= A sin x + B cos x , then prove that (d^(2)y)/(dx^(2)) +y= 0