Question

If $S_n = \cos^n\, \theta + \sin^n\, \theta $ , then the value of $ 3S_4 − 2S_6 $ is given by

• A. 4
• B. 0
• C. 1
• D. 7

Answer 1

Answer: (C)

Given, $S_{n}=\cos ^{n} \theta+\sin ^{n} \theta$
$\therefore 3 S_{4}-2 S_{6}=3\left[\left(\cos ^{4} \theta+\sin ^{4} \theta\right)\right]-2\left[\cos ^{6} \theta+\sin ^{6} \theta\right]$
$=3\left[\left(\cos ^{2} \theta+\sin ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cos ^{2} \theta\right]$
$-2\left[\left(\cos ^{2} \theta+\sin ^{2} \theta\right)\left(\cos ^{4} \theta+\sin ^{4} \theta\right.\right.\left.\left.-\cos ^{2} \theta \,\sin ^{2} \theta\right)\right]$
$=3\left[1-2 \sin ^{2} \theta \,\cos ^{2} \theta\right]-2\left[\left(\cos ^{2} \theta+\sin ^{2} \theta\right)^{2}\right.\left.-3 \cos ^{2} \theta \sin ^{2} \theta\right]$
$=3-6 \sin ^{2} \theta \,\cos ^{2} \theta-2+6 \cos ^{2} \theta \,\sin ^{2} \theta$
$=1$