Question

solve: sin4theta+sin2theta=costheta​

Answer 1

Step-by-step explanation:

We have,

$\sin(4 \theta) + \sin(2 \theta) = \cos( \theta)$

$\implies \: 2\sin \bigg( \frac{4 \theta + 2 \theta}{2} \bigg) \cos \bigg( \frac{4 \theta - 2 \theta}{2} \bigg) = \cos( \theta)$

$\implies \: 2\sin ( 3 \theta ) \cos ( \theta ) = \cos( \theta)$

$\implies \: 2\sin ( 3 \theta ) \cos ( \theta ) - \cos( \theta) = 0$

$\implies \cos ( \theta )(2\sin ( 3 \theta ) - 1) = 0$

$\implies \cos ( \theta ) = 0 \: \: or \: \: 2\sin ( 3 \theta ) - 1 = 0$

$\implies \cos ( \theta ) = 0 \: \: or \: \: \sin ( 3 \theta ) = \frac{1}{2}$

$\implies \theta = (2n + 1) \frac{\pi}{2} \: \: or \: \: 3 \theta = m\pi + ( - 1) ^{m}. \frac{\pi}{6}$

$\implies \theta = (2n + 1) \frac{\pi}{2} \: \: or \: \: \theta = \frac{m\pi }{3} + ( - 1) ^{m}. \frac{\pi}{18}$