What is the general solution for (1+csc(γ))/(cot(γ)+cos(γ))=csc(γ) ?

The general solution for (1+csc(γ))/(cot(γ)+cos(γ))=csc(γ) is γ= pi/4+pin

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(1+csc(γ))/(cot(γ)+cos(γ))=csc(γ)

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} = csc (γ)

γ = \frac{π}{4} + πn

Solution steps

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} = csc (γ)

Subtract

csc (γ)

from both sides

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} - csc (γ) = 0

Simplify

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} - csc (γ) : \frac{1 + csc ( γ ) - csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )}

\frac{1 + csc ( γ ) - csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )} = 0

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

1 + csc (γ) - csc (γ) (cot (γ) + cos (γ)) = 0

Express with sin, cos

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

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

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

Factor

- cos (γ) + sin (γ) + sin^{2} (γ) - cos (γ) sin (γ) : (1 + sin (γ)) (- cos (γ) + sin (γ))

(1 + sin (γ)) (- cos (γ) + sin (γ)) = 0

Solving each part separately

1 + sin (γ) = 0 or - cos (γ) + sin (γ) = 0

1 + sin (γ) = 0 : γ = \frac{3 π}{2} + 2 πn

- cos (γ) + sin (γ) = 0 : γ = \frac{π}{4} + πn

Combine all the solutions

γ = \frac{3 π}{2} + 2 πn, γ = \frac{π}{4} + πn

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

γ = \frac{π}{4} + πn

sin (θ) = \frac{2}{6}

3 sin (2 x) = - 2

x, arctan (y) = 2 arctan (x)

sin (2 x) = - \frac{3}{2}

tan (x) = \frac{7.3}{6.8}

What is the general solution for (1+csc(γ))/(cot(γ)+cos(γ))=csc(γ) ?
The general solution for (1+csc(γ))/(cot(γ)+cos(γ))=csc(γ) is γ= pi/4+pin