# If x = a sin θ and y = b cos θ, what is the value of b2x2 + a2y2?

Question:

If $x=a \sin \theta$ and $y=b \cos \theta$, what is the value of $b^{2} x^{2}+a^{2} y^{2}$ ?

Solution:

Given:

$x=a \sin \theta$ and $y=b \cos \theta$

So,

$b^{2} x^{2}+a^{2} y^{2}=b^{2}(a \sin \theta)^{2}+a^{2}(b \cos \theta)^{2}$

$=a^{2} b^{2} \sin ^{2} \theta+a^{2} b^{2} \cos ^{2} \theta$

$=a^{2} b^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)$

We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$

Therefore, $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}$