What is the general solution for cot(x)csc(x)=cos(x) ?

The general solution for cot(x)csc(x)=cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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cot(x)csc(x)=cos(x)

cot (x) csc (x) = cos (x)

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

Solution steps

cot (x) csc (x) = cos (x)

Subtract

cos (x)

from both sides

cot (x) csc (x) - cos (x) = 0

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

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

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

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

Combine all the solutions

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

2 tan (x) = 2

0.5 = cos (\frac{π}{6} x)

cos^{2} (x) = \frac{π}{4}

cot^{2} (θ) + csc (θ) = 1, 0 \leq θ < 2 π

tan (x) = \frac{sec ( x )}{cos ( x )}

What is the general solution for cot(x)csc(x)=cos(x) ?
The general solution for cot(x)csc(x)=cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin