If `a^(2)sec^(2)alpha-b^(2)tan^(2)alpha=c^(2)`, then prove that `sinalpha=pm sqrt((c^(2)-a^(2))/(c^(2)-b^(2)))`.

If cos^2alpha-sin^2alpha=tan^2beta," then prove that "tan^2alpha=cos^2beta-sin^2beta .

If alpha+beta=(pi)/(2) , then prove that cosalpha=sqrt((sinalpha)/(cosbeta)-sinalphacosbeta)

If a : b = c : d , then prove that (a + c) : (b + d) = sqrt(a^(2) - c^(2)) : sqrt(b^(2) - d^(2))

x=sqrt(a^(2)cos^(2)alpha+b^(2)sin^(2)alpha)+sqrt(alpha^(2)sin^(2)alpha+b^(2)cos^(2)alpha) then x^(2)=alpha^(2)+b^(2)+2sqrt(p(a^(2)+b^(2))-p^(2)) , where p is equal to

If a sin x=b cosx=(2c tan x)/(1-tan^(2)x) , then prove that (a^(2)-b^(2))^(2)=4c^(2)(a^(2)+b^(2)).

If sinalpha+sinbeta=a and cosalpha+cosbeta=b , prove that tan((alpha-beta)/2)=+-sqrt((4-a^2-b^2)/(a^2+b^2)) .

sinalpha+sinbeta=a and cosalpha+cosbeta=b prove that , tan((alpha-beta)/2)=+-sqrt((4-a^2-b^2)/(a^2+b^2))

If angles alpha and beta satisfy the equation a cos theta+ b sin theta= c(a,b,c are constants), prove that- (a) sin (alpha+ beta)= (2ab)/(a^(2)+b^(2)) (b) cos (alpha+beta)=(a^(2)-b^(2))/(a^(2)+b^(2)) (c) cos (alpha- beta)=(2c^(2)-(a^(2)+b^(2)))/(a^(2)+b^(2))

If the line x cos alpha+ y sin alpha=p be a normal to the hyperbola b^(2)x^(2)-a^(2)y^(2)=a^(2)b^(2), show that, p^(2)(a^(2) sec ^(2) alpha- b^(2)"cosec"^(2) alpha)=(a^(2)+b^(2))^(2)

If alpha and beta be two roots of the equation a cos theta+ b sin theta=c , show that sin alpha+ sin beta=(2bc)/(a^(2)+b^(2)) ,sin alpha sin beta =(c^(2)-a^(2))/(a^(2)+b^(2)) and tan (alpha+ beta)=(2ab)/(a^(2)-b^(2))