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

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

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(sec(x)+tan(x))/(cot(x)+cos(x))=sec^2(x)

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

No Solution for x \in R

Solution steps

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

Subtract

sec^{2} (x)

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

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

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

- 1 + sin^{2} (x) = 0

Solve by substitution

sin (x) = 1, sin (x) = - 1

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

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

No Solution for x \in R

cos (x) = csc (x)

1 + cos (2 t) = cos^{2} (t)

2 sin (2 x + \frac{3 π}{2}) + 1 = 0

2 cos (2 x) - sin (x) - 1 = 0

2 cos^{2} (θ) - 2 sin^{2} (θ) = 2

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