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

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

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(1+csc(y))/(cot(y)+cos(y))=csc(y)

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

y = \frac{π}{4} + πn

Solution steps

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

Subtract

csc (y)

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

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

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

Combine all the solutions

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

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

y = \frac{π}{4} + πn

2 = sin (10 x)

cos (2 x) = \frac{( 2 k - 8 )}{9}

x, h = tan (x - \frac{π}{4})

cos^{3} (x) + sin (x) = 0

12 cos^{2} (θ) + 5 sin (θ) - 9 = 0

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