Question

Solve the equation frac{(2sin theta sin 2theta)}{cos theta+cos 3theta}=tan (theta) tan(2theta)

Trigonometric equation and identitie
ANSWERED
asked 2020-11-12
Solve the equation \(\frac{(2\sin \theta \sin 2\theta)}{\cos \theta+\cos 3\theta}=\tan (\theta) \tan(2\theta)\)

Answers (1)

2020-11-13

\(\frac{(2\sin \theta \sin2\theta)}{\cos \theta+\cos3\theta}\)
\(=\frac{2\sin(\theta)\sin(2\theta)}{-2\cos(\theta)+4\cos^{3}(\theta)}\)
\(=\frac{2\sin(\theta)\sin(2\theta)}{2\cos(\theta)(-1+\cos^{2}(\theta))}\)
\(=\frac{2\sin(\theta)\sin(2\theta)}{\cos(\theta)(2\cos^2(\theta)-1)}\)
Use the following identity: \(\frac{\sin(x)}{\cos(x)}=\tan(x)\)
\(=\frac{\sin(2\theta)\tan(\theta)}{(2\cos^2(\theta)-1)}\)
\(=\frac{\sin(2\theta)\tan(\theta)}{\cos(2\theta}\)
Use the following identity: \(\frac{\sin(x)}{\cos(x)}=\tan(x)\)
\(=\frac{\tan(\theta)}{tan(2\theta}\)
Therefore
\(\frac{2\sin \theta \sin2\theta}{\cos \theta+\cos3\theta}=\tan (\theta) \tan(2\theta)\)

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