Question

If $cos5\theta =5cos\theta -20co{s}^3\theta +aco{s}^5\theta +b$ , then the value of $a+b$ is equal to

• A. $20$
• B. $16$
• C. $-16$
• D. $15$

Answer 1

Answer: (B)

$cos5\theta =cos\left(4\theta +\theta \right)$
$=cos4\theta cos\theta -sin4\theta sin\theta$
$=\left(2{\left(\left[2co{s}^2\theta -1\right]\right)}^2-1\right)cos\theta -2.2sin\theta cos\theta \left[2co{s}^2\theta -1\right]sin\theta$
$=cos\theta \left[2\left(4{\left(cos\right)}^4\theta +1-4{\left(cos\right)}^2\theta \right)-1\right]$ $-4si{n}^2\theta cos\theta \left[2co{s}^2\theta -1\right]$
$=8co{s}^5\theta -8co{s}^3\theta +2cos\theta -cos\theta$ $-4\left(1-{\left(cos\right)}^2\theta \right)cos\theta \left(2{\left(cos\right)}^2\theta -1\right)$
$=8co{s}^5\theta -8co{s}^3\theta +cos\theta -4cos\theta$ $\left[2co{s}^2\theta -1-2co{s}^4\theta +co{s}^2\theta \right]$
$=16co{s}^5\theta -20co{s}^3\theta +5cos\theta$
Clearly, $a=16,b=0$