What is the general solution for (cot(a))/(cos(a))=cos(a) ?

The general solution for (cot(a))/(cos(a))=cos(a) is No Solution for a\in\mathbb{R}

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(cot(a))/(cos(a))=cos(a)

\frac{cot ( a )}{cos ( a )} = cos (a)

No Solution for a \in R

Solution steps

\frac{cot ( a )}{cos ( a )} = cos (a)

Subtract

cos (a)

from both sides

\frac{cot ( a )}{cos ( a )} - cos (a) = 0

Simplify

\frac{cot ( a )}{cos ( a )} - cos (a) : \frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )}

\frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )} = 0

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

cot (a) - cos^{2} (a) = 0

Express with sin, cos

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

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

cos (a) - cos^{2} (a) sin (a) = 0

Factor

cos (a) - cos^{2} (a) sin (a) : cos (a) (1 - sin (a) cos (a))

cos (a) (1 - sin (a) cos (a)) = 0

Solving each part separately

cos (a) = 0 or 1 - sin (a) cos (a) = 0

cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

1 - sin (a) cos (a) = 0 :

No Solution

Combine all the solutions

a = \frac{π}{2} + 2 πn, a = \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 a \in R

cos (x) - 2 = 0

tan^{2} (x) + 3 = 3 sec (x)

3 cot^{2} (x) - 4 cot (x) + 3 = 0

cot (θ) + tan (θ) = 2

3 cos^{2} (θ) - 1 = 0

What is the general solution for (cot(a))/(cos(a))=cos(a) ?
The general solution for (cot(a))/(cos(a))=cos(a) is No Solution for a\in\mathbb{R}