What is the general solution for (sec(x)sin(x))/(tan(x)cot(x))=sin^2(x) ?

The general solution for (sec(x)sin(x))/(tan(x)cot(x))=sin^2(x) is No Solution for x\in\mathbb{R}

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(sec(x)sin(x))/(tan(x)cot(x))=sin^2(x)

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

No Solution for x \in R

Solution steps

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

Subtract

sin^{2} (x)

from both sides

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) = 0

Simplify

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) : \frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )} = 0

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

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x) = 0

Factor

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x) : sin (x) (sec (x) - sin (x) tan (x) cot (x))

sin (x) (sec (x) - sin (x) tan (x) cot (x)) = 0

Solving each part separately

sin (x) = 0 or sec (x) - sin (x) tan (x) cot (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sec (x) - sin (x) tan (x) cot (x) = 0 :

No Solution

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

tan (θ) = - \frac{3}{1}

sin (3 x) + cos (3 x) = 0

sin (x) = - \frac{3}{4}, π < x < \frac{3 π}{4}

cos (2 x) + cos (x) = 0, 0 \leq x < 2 π

10 cos^{2} (x) = 5

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