A line `x cos alpha + y sin alpha = P` is a tangent to the curve `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`. Prove that `a^(2)cos^(2)alpha+b^(2)sin^(2)alpha=P^(2)`.

If the straight line x cos alpha + y sin alpha = p touches the curve (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 then prove that a^(2)cos^(2)alpha+b^(2)sin^(2)alpha= p^(2) .

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

If sin (3alpha) /cos(2alpha) < 0 If alpha lies in

If y= sin (pt), x= sin t , then prove that (1- x^(2))y_(2)-xy_(1)+ p^(2)y= 0

If cos^(-1)x/a+cos^(-1)y/b=alpha, prove that (x^2)/(a^2)-2(x y)/(a b)cosalpha+(y^2)/(b^2)=sin^2alpha

The angle between the lines (x^(2)+y^(2))sin^(2)alpha=(x cos beta-y sin beta)^(2) is

If 3 sin alpha=5 sin beta , then (tan((alpha+beta)/2))/(tan ((alpha-beta)/2))=

If Delta_(1) is the area of the triangle with vertices (0, 0), (a tan alpha, b cot alpha), (a sin alpha, b cos alpha), Delta_(2) is the area of the triangle with vertices (a sec^(2) alpha, b cos ec^(2) alpha), (a + a sin^(2)alpha, b + b cos^(2)alpha) and Delta_(3) is the area of the triangle with vertices (0,0), (a tan alpha, -b cot alpha), (a sin alpha, b cos alpha) . Show that there is no value of alpha for which Delta_(1), Delta_(2) and Delta_(3) are in GP.

The greatest & the least magnitude of the resultant of two forces of constant magnitudes are P & Q respectively. If the forces act at an angle 2 alpha , then prove that the magnitude of their resultant is given by sqrt(P^(2) cos^(2) alpha+Q^(2) sin^(2) alpha) .