Question

If x sin^3 theta + y cos^3 theta = sin theta cos theta and x sin theta = y cos theta ; prove that: x^2 + y^2 = 1

Answer 1

$\textbf{Given:}$

$x\,sin\,\theta=y\,cos\,\theta$

$\implies\,\frac{sin\,\theta}{cos\,\theta}=\frac{y}{x}$

$\text{Then, }sin\,\theta=k\,y\;\;\&\;\;cos\,\theta=k\,x$

$\text{Using, }\bf\,sin^2\theta+cos^2\theta=1$

$\implies\,k^2y^2+k^2x^2=1$

$\implies\,k^2(y^2+x^2)=1$

$\implies\bf\,k=\frac{1}{\sqrt{x^2+y^2}}$

$\text{Also, }x\,sin^3\theta+y\,cos^3 \theta=sin\theta \,cos\theta$

$\implies\,x\,(k\,y)^3+y\,(k\,x)^3 =(k\,y)(k\,x)$

$\implies\,x\,k^3y^3+y\,k^3x^3 =k^2x\,y$

$\implies\,k^3(x\,y^3+y\,x^3) =k^2x\,y$

$\implies\,k\,x\,y(y^2+x^2) =x\,y$

$\implies\,k(y^2+x^2) =1$

$\implies\frac{1}{\sqrt{x^2+y^2}}(y^2+x^2) =1$

$\implies\sqrt{x^2+y^2}=1$

$\text{Squaring on both sides, we get}$

$\implies\bf\,x^2+y^2=1$

Answer 2

Answer:

Explanation in attachment.