Question

solve sinxsin3x = 1/2​

Answer 1

sinxsin3x = 1/2

2sinxsin3x = 1

cos-2x - cos4x = 1 (2sinasinb = cosa-b - cosa+b)

cos2x - cos4x = 1 ...........1

cos4x = 2cos22x - 1 so eq 1 becomes

cos2x - (2cos22x-1) = 1

cos2x - 2cos22x = 0

cos2x(1-2cos2x) = 0

cos2x = 0 or 1-2cos2x = 0

2x = (2n+1)pi/2 2x = 2npi +(-)pi/3

x = (2n+1)pi/4 or x = npi +(-)pi/6

Answer 2

$2 \sin(3x) \sin(x) = 1 \\ \cos(3x - x) - \cos(3x + x) = 1 \\ \cos(2x) - \cos(4x) = 1 \\ \cos(2x) -( 2 ({ \cos(2x)) }^{2} - 1) = 1 \\ \\ let \: \cos(2x) = p \\ 2 {p}^{2} - p = 0 \\ p(2p - 1) = 0 \\ \cos(2x) = 0 \: \: \: \: \: \: \: \: \: \: \: \: or \: \: \cos(2x) = \frac{1}{2} \\ 2x = \frac{\pi}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: or \: \: \: 2x = \frac{\pi}{3} \\ x = \frac{\pi}{4} \: \: or \: \: \frac{\pi}{6}$

here we r taking only principle values