What is the general solution for (sec(B)+csc(B))/(1+tan(B))=csc^2(B) ?

The general solution for (sec(B)+csc(B))/(1+tan(B))=csc^2(B) is No Solution for B\in\mathbb{R}

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(sec(B)+csc(B))/(1+tan(B))=csc^2(B)

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} = csc^{2} (B)

No Solution for B \in R

Solution steps

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} = csc^{2} (B)

Subtract

csc^{2} (B)

from both sides

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} - csc^{2} (B) = 0

Simplify

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} - csc^{2} (B) : \frac{sec ( B ) + csc ( B ) - csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )}

\frac{sec ( B ) + csc ( B ) - csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )} = 0

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

sec (B) + csc (B) - csc^{2} (B) (1 + tan (B)) = 0

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

sin (B) + cos (B) = 0 : B = \frac{3 π}{4} + πn

sin (B) - 1 = 0 : B = \frac{π}{2} + 2 πn

Combine all the solutions

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

Since the equation is undefined for:

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

No Solution for B \in R

4 = 2 csc (3 x)

21 = 24 + 8 cos (\frac{π x}{6})

sin (x) = \frac{16 sin ( 3 1 ^{\circ} )}{12}

sinh (z) - cosh (z) = 0

cos (x) = - 1.588908648

What is the general solution for (sec(B)+csc(B))/(1+tan(B))=csc^2(B) ?
The general solution for (sec(B)+csc(B))/(1+tan(B))=csc^2(B) is No Solution for B\in\mathbb{R}