`P rov et h a t1

If y=(sinx)/(1+(cosx)/(1+(sinx)/(1+(cosx)/(1+ tooo)))),p rov et h a t(dy)/(dx)=((1+y)cosx+ysinx)/(1+2y+cosx-sinx)

If y=sqrt(sinx+sqrt(sinx+sqrt(sinx+ tooo))),p rov et h a t(dy)/(dx)=(cosx)/(2y-1)

If x=cos t and y=sin t,p ro v et h a t(dy)/(dx)=(1)/(sqrt(3))att=(2 pi)/(3)

If sqrt(1-x^6)+sqrt(1-y^6)=a(x^3-y^3),p rov et h a t(dy)/(dx)=(x^2)/(y^2)sqrt((1-y^6)/(1-x^6,) w h e r e-1x,1a n d-1

If x=a(1-cos^3theta),y=asin^3theta,"p r o v et h a t"(d^2y)/(dx^2)=(32)/(27a) "at" theta=pi/6dot

y sqrt(1-x^(2))+x sqrt(1-y^(2))=1,prov et h a t (dy)/(dx)=sqrt((1-y^(2))/(1-x^(2)))

x=a sec theta,y=b tan theta,pr@v et h a t(d^(2)y)/(dx^(2))=-(b^(4))/(a^(2)y^(3))

In an A.P., if p^(t h) term is 1/q and q^(t h) term is 1/p , prove that the sum of first pq terms is 1/2(p q+1), where p!=q .

In a G.P. it is given that T_(p-1)+T_(p+1)=3T_(p). Prove that its common ratio is an irrational number.