Question

If $\sin x+\cos x=\frac{1}{2}$ then ${\sin }^{4}x+{\cos }^{4}x$ as a rational number equals

• A. $\frac{3}{4}$
• B. $\frac{15}{32}$
• C. $\frac{19}{32}$
• D. $\frac{23}{32}$

Answer 1

Answer: (D)

$\frac{23}{32}$

$\sin x+\cos x=\frac{1}{2}$
Squaring both sides,
${\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x=\frac{1}{4}$
$1+2\sin x\cos x=\frac{1}{4}$
$\sin x\cos x=\frac{-3}{8}$..(i)

$({\sin }^{2}x+{\cos }^{2}x)({\sin }^{2}x+{\cos }^{2}x)=1$
${\sin }^{4}x+{\cos }^{4}x+2{\sin }^{2}x{\cos }^{2}x=1$
${\sin }^{4}x+{\cos }^{4}x+2{\left(\frac{-3}{8}\right)}^2=1$...From (i)
${\sin }^{4}x+{\cos }^{4}x+\frac{9}{32}=1$
${\sin }^{4}x+{\cos }^{4}x=1-\frac{9}{32}$
${\sin }^{4}x+{\cos }^{4}x=\frac{23}{32}$