Question

Period of $2{\sin }^{3}x+5{\cos }^{3}x$ is $2\pi$
II Period of ${\sin }^{4}x+{\cos }^{4}x$ is $\frac{\pi }{2}$

• A. only I is true
• B. only II is true
• C. Both I and II are true
• D. Neither I nor II true

Answer 1

Answer: (C)

Both I and II are true

I. $f\left(x\right)=2{\sin }^{3}x+5{\cos }^{3}x$
$\cos 3x=4{\cos }^{3}x-3\cos x$
$\sin 3x=3\sin x-4{\sin }^{3}x$
$f\left(x\right)=\frac{5}{4}(\cos 3x+3\cos x)+\frac{(3\sin x-\sin 3x)}{2}$
$=\frac{5}{4}\cos 3x-\frac{1}{2}\sin 3x+\frac{15}{4}\cos x+\frac{3}{2}\sin x$
$\therefore$ Period of $f\left(x\right)=LCMof(\frac{2\pi }{3},\frac{2\pi }{3},2\pi ,2\pi )$
$=2\pi$ True.
II. $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$
$=1-\frac{(\sin 2x{)}^2}{2}=\frac{1-\left(\frac{1-\cos 4x}{2}\right)}{2}=\frac{1+\cos 4x}{4}$
Period of $f\left(x\right)=$ period of $\cos 4x=\frac{2\pi }{4}=\frac{\pi }{2}$. True $[\cos 2x=1-2{\sin }^{2}x]$