Question

Find the value of ${\cos }^6\Theta +{\sin }^6\Theta$.

• A. $\frac{1}{8}(5-3\cos 4\Theta +\cos 6\Theta )$
• B. $\frac{1}{16}(10-\cos 4\Theta +\cos 6\Theta )$
• C. $\frac{1}{8}(5+3\cos 4\Theta )$
• D. $\frac{1}{16}(10+\cos 4\Theta )$

Answer 1

Answer: (C)

$\frac{1}{8}(5+3\cos 4\Theta )$

Take $a={\cos }^2\Theta$ and $b={\sin }^2\Theta$
given expression is
$a^3+b^3=(a+b)(a^2-ab+b^2)$
$=(a+b)\left(\right(a+b{)}^2-3ab)$
but a+b = 1
so given expression becomes
$=1-3ab$
$=1-3{\sin }^2\Theta {\cos }^2\Theta$
$=1-\frac{3}{4}{\sin }^{2}2\Theta$
$=\frac{1}{8}(5+3\cos 4\Theta )$