If `xsin^3theta+ycos^3theta-sinthetacosthetaa n dxsintheta=ycostheta,`

1.	If `xsin^3theta+ycos^3theta-sinthetacosthetaa n dxsintheta=ycostheta,`prove that `x^2+y^2=1`
Answer» ` xsin^(3) theta + y cos^(3) theta = sin theta cos theta ` ` rArr (xsin theta)sin^(2)theta + (ycos theta )cos^(2)theta = sin theta cos theta ` ` rArr (xsin theta)sin^(2)theta + (xsin theta)cos^(2)theta = sin theta cos theta " " [because y cos theta = x sin theta ] ` `rArr (xsin theta ) (sin^(2) theta + cos^(2) theta ) = sin theta cos theta ` ` rArr xsin theta = sin theta cos theta " " [ because sin^(2) theta + cos^(2) theta =1 ] ` ` rArr x= cos theta . " "...(i) ` Now ,`xsin theta = y cos theta ` `rArr cos theta sin theta = y cos theta " "[because x= cos theta] ` ` rArr y= sin theta. " "...(ii) ` On squaring (i) and (ii) and adding, we get ` x^(2) + y^(2) =1. ` Hence, ` x^(2) + y^(2) =1. `

Discussion

`1/(c o s e ctheta-cottheta)-1/(sintheta)=1/(sintheta)-1/(c o s e ctheta+cottheta)`
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