What is the general solution for 1/(tan(α))+tan(α)= 1/(sin(α)) ?

The general solution for 1/(tan(α))+tan(α)= 1/(sin(α)) is No Solution for α\in\mathbb{R}

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1/(tan(α))+tan(α)= 1/(sin(α))

\frac{1}{tan ( α )} + tan (α) = \frac{1}{sin ( α )}

No Solution for α \in R

Solution steps

\frac{1}{tan ( α )} + tan (α) = \frac{1}{sin ( α )}

Subtract

\frac{1}{sin ( α )}

from both sides

\frac{1}{tan ( α )} + tan (α) - \frac{1}{sin ( α )} = 0

Simplify

\frac{1}{tan ( α )} + tan (α) - \frac{1}{sin ( α )} : \frac{sin ( α ) + tan ^{2} ( α ) sin ( α ) - tan ( α )}{tan ( α ) sin ( α )}

\frac{sin ( α ) + tan ^{2} ( α ) sin ( α ) - tan ( α )}{tan ( α ) sin ( α )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (α) + tan^{2} (α) sin (α) - tan (α) = 0

Express with sin, cos

\frac{sin ^{3} ( α ) - cos ( α ) sin ( α ) + cos ^{2} ( α ) sin ( α )}{cos ^{2} ( α )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin^{3} (α) - cos (α) sin (α) + cos^{2} (α) sin (α) = 0

Factor

sin^{3} (α) - cos (α) sin (α) + cos^{2} (α) sin (α) : sin (α) (sin^{2} (α) - cos (α) + cos^{2} (α))

sin (α) (sin^{2} (α) - cos (α) + cos^{2} (α)) = 0

Rewrite using trig identities

sin (α) (- cos (α) + 1) = 0

Solving each part separately

sin (α) = 0 or - cos (α) + 1 = 0

sin (α) = 0 : α = 2 πn, α = π + 2 πn

- cos (α) + 1 = 0 : α = 2 πn

Combine all the solutions

α = 2 πn, α = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for α \in R

tan (2 θ) = \frac{- 81}{81}

sin (θ) cos (2 θ) - cos (2 θ) = 0

tan (θ) = - \frac{7}{6}

8 sin^{2} (x) - 1 = 5

sin (d) = 0.8

What is the general solution for 1/(tan(α))+tan(α)= 1/(sin(α)) ?
The general solution for 1/(tan(α))+tan(α)= 1/(sin(α)) is No Solution for α\in\mathbb{R}