Question

If $a{\sin }^{3}x+b{\cos }^{3}x=\sin x\cos x$ and $a\sin x=b\cos x$ . If $a,b,\sin x$ and $\cos x$ all are non zero real numbers, then $a^2+b^2=$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$1$

Given,
$a{\sin }^{3}x+b{\cos }^{3}x=\sin x\cos x...........\left(1\right)$

$a\sin x=b\cos x.........\left(2\right)$

Consider equation $\left(1\right)$,

$a{\sin }^{3}x+b{\cos }^{3}x=\sin x\cos x$

$a\sin x{\sin }^{2}x+b{\cos }^{3}x=\sin x\cos xb\cos x{\sin }^{2}x+b{\cos }^{3}x=\sin x\cos xb\cos x({\sin }^{2}x+{\cos }^{2}x)=\sin x\cos x$ (using $\left(2\right)$)

Solving this, we get

$b=\sin x$

Then $a=\frac{b\cos x}{\sin x}=\cos x$

Hence,

$a^2+b^2={\sin }^{2}x+{\cos }^{2}x=1$