What is the general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 ?

The general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 is No Solution for x\in\mathbb{R}

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1/(tan(x))+(cos^2(x))/(sin(x))=0

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} = 0

No Solution for x \in R

Solution steps

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} = 0

Simplify

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} : \frac{sin ( x ) + cos ^{2} ( x ) tan ( x )}{tan ( x ) sin ( x )}

\frac{sin ( x ) + cos ^{2} ( x ) tan ( x )}{tan ( x ) sin ( x )} = 0

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

sin (x) + cos^{2} (x) tan (x) = 0

Rewrite using trig identities

sin (x) + cos (x) sin (x) = 0

Factor

sin (x) + cos (x) sin (x) : sin (x) (cos (x) + 1)

sin (x) (cos (x) + 1) = 0

Solving each part separately

sin (x) = 0 or cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

1 sin (2 5^{\circ}) = 1.506 sin (x)

\frac{1}{2} tan (2 θ) = tan (θ)

csc (x) - 3 = - 2 sin (x)

\frac{sin ( y )}{80} = \frac{sin ( 4 0 ^{\circ} )}{60}

cos (w) = 0.75 sin (9 0^{\circ} - w)

What is the general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 ?
The general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 is No Solution for x\in\mathbb{R}