Question

Find the value of ${\cos }^5\theta .$

• A. $\frac{1}{16}(\cos 5\theta +5\cos 3\theta +10\cos \theta )$
• B. $\frac{1}{16}(\cos 5\theta -5\cos 3\theta +10\cos \theta )$
• C. $\frac{1}{32}(\cos 5\theta -5\cos 3\theta +10\cos \theta )$
• D. $\frac{1}{32}(\cos 5\theta +5\cos 3\theta +10\cos \theta )$

Answer 1

Answer: (A)

$\frac{1}{16}(\cos 5\theta +5\cos 3\theta +10\cos \theta )$

Let

$x=\cos \theta +i\sin \theta$ and $x^n=\cos n\theta +i\sin n\theta$.....(By De movier's theorem)

$\frac{1}{x}=\cos \theta -i\sin \theta$

Therefore

$x+\frac{1}{x}=2\cos \theta$

And

$x^n+{\frac{1}{x}}^n=2\cos n\theta$

Therefore

$(x+\frac{1}{x}{)}^5=32{\cos }^5\theta$

$x^5+5x^3+10x+\frac{10}{x}+\frac{5}{x^3}+\frac{1}{x^5}=32{\cos }^5\theta$

$(x^5+\frac{1}{x^5})+5(x^3+\frac{1}{x^3})+10(x+\frac{1}{x})=32{\cos }^5\theta$

$2[\cos 5\theta +5\cos 3\theta +10\cos \theta ]=32{\cos }^5\theta$

$\frac{1}{16}[\cos 5\theta +5\cos 3\theta +10\cos \theta )={\cos }^5\theta$.