Solve the equation 1csc⁡θ−cos⁡θ=1+cos⁡θsin⁡θ

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gotovub · Accepted Answer

We work using the left side. Write in terms of sine and cosine:
$\frac{1}{\csc θ - \cos θ} = \frac{1}{\frac{1}{\sin θ} - \frac{\cos θ}{\sin θ}}$
$\frac{1}{\csc θ - \cos θ} = \frac{1}{\frac{1 - \cos θ}{\sin θ}}$
$\frac{1}{\csc θ - \cos θ} = \frac{\sin θ}{1 - \cos θ}$
Multiply by $\frac{1 + \cos θ}{1 + \cos θ}$ :
$\frac{1}{\csc θ - \cos θ} = \frac{\sin}{1 - \cos θ} \times \frac{1 + \cos θ}{1 + \cos θ}$
$\frac{1}{\csc θ - \cos θ} = \frac{\sin θ (1 + \cos θ)}{1 - \cos^{2} θ}$
Use the Pythagorean identity: $\sin^{2} θ + \cos^{2} θ = 1 and \sin^{2} θ = 1 - \cos^{2} θ$
$\frac{1}{\csc θ - \cos θ} = \frac{\sin θ (1 + \cos θ)}{\sin^{2} θ}$
$\frac{1}{\csc θ - \cos θ} = \frac{1 + \cos θ}{\sin θ}$

Solve the equation frac{1}{csc theta-cos theta}=frac{1+cos theta}{sin theta}

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