Best solutions to - "Solve the equation (2sin⁡θsin⁡2θ)cos⁡θ+cos⁡3θ=tan⁡(θ)tan⁡(2θ)"

High School
Geometry
Trigonometry
Trigonometric equations and identities

illusiia

Answered question

2020-11-12

Solve the equation

\frac{(2 \sin θ \sin 2 θ)}{\cos θ + \cos 3 θ} = \tan (θ) \tan (2 θ)

Answer & Explanation

tafzijdeq

Skilled2020-11-13Added 92 answers

$\frac{(2 \sin θ \sin 2 θ)}{\cos θ + \cos 3 θ}$
$= \frac{2 \sin (θ) \sin (2 θ)}{- 2 \cos (θ) + 4 \cos^{3} (θ)}$
$= \frac{2 \sin (θ) \sin (2 θ)}{2 \cos (θ) (- 1 + \cos^{2} (θ))}$
$= \frac{2 \sin (θ) \sin (2 θ)}{\cos (θ) (2 \cos^{2} (θ) - 1)}$
Use the following identity: $\frac{\sin (x)}{\cos (x)} = \tan (x)$
$= \frac{\sin (2 θ) \tan (θ)}{(2 \cos^{2} (θ) - 1)}$
$= \frac{\sin (2 θ) \tan (θ)}{\cos (2 θ}$
Use the following identity: $\frac{\sin (x)}{\cos (x)} = \tan (x)$
$= \frac{\tan (θ)}{t a n (2 θ}$
Therefore
$\frac{2 \sin θ \sin 2 θ}{\cos θ + \cos 3 θ} = \tan (θ) \tan (2 θ)$

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