If `p^(2)=a^(2)cos^(2)theta+b^(2)sin^(2)theta`, show that, `p+(d^(2)p)/(d theta^(2))=(a^(2)b^(2))/(p^(3))`.

If P=(1)/(2)sin^(2)theta+(1)/(3)cos^(2)theta then

int_(0)^((pi)/(2))sinthetacostheta(a^(2)sin^(2)theta+b^(2)cos^(2)theta)^((1)/(2))d theta=(1)/(3)((a^(2)+ab+b^(2))/(a+b))

If u=sqrt(a ^(2) cos ^(2) theta+b ^(2) sin ^(2) theta)+ sqrt( a^(2) sin ^(2) theta+b ^(2) cos ^(2) theta), then the difference between the maximum and minimum values of u^(2) is-

int (sin^(2)theta d theta)/((1 + cos theta)^(2))

If x+a= a(2 cos theta-cos 2 theta) and y= a(2 sin theta- sin 2 theta) ,show that (x^(2)+y^(2)+2ax)^(2)=4a^(2)(x^(2)+y^(2)) .

If sin theta+sin^2theta=1 , show that cos^4theta+cos^2theta=1 .

If sin theta + sin 2 theta = a " and " cos theta + cos 2 theta = b, "show that," (a^(2) + b^(2))(a^(2) +b^(2) -3)=2b

If a^(2)(1- sin theta) +b^(2)(1+ sin theta)= 2 ab cos theta then show that 2tan theta=(a)/(b)-(a)/(a)

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

int_(0)^((pi)/(2))(sin^(3)theta)/(sin^(3)theta+cos^(3)theta)d theta