What is the general solution for (cos^3(x))/(sin(x))=cot(x) ?

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

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(cos^3(x))/(sin(x))=cot(x)

\frac{cos ^{3} ( x )}{sin ( x )} = cot (x)

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

Solution steps

\frac{cos ^{3} ( x )}{sin ( x )} = cot (x)

Subtract

cot (x)

from both sides

\frac{cos ^{3} ( x )}{sin ( x )} - cot (x) = 0

Simplify

\frac{cos ^{3} ( x )}{sin ( x )} - cot (x) : \frac{cos ^{3} ( x ) - cot ( x ) sin ( x )}{sin ( x )}

\frac{cos ^{3} ( x ) - cot ( x ) sin ( x )}{sin ( x )} = 0

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

cos^{3} (x) - cot (x) sin (x) = 0

Rewrite using trig identities

- cos (x) + cos^{3} (x) = 0

Solve by substitution

cos (x) = 0, cos (x) = - 1, cos (x) = 1

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

cos (x) = - 1 : x = π + 2 πn

cos (x) = 1 : x = 2 πn

Combine all the solutions

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

Since the equation is undefined for:

π + 2 πn, 2 πn

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

tan (2 x) = \frac{3}{4}

sin (x) = 1 + cos (x)

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

sin (x) = \frac{21}{5}

tan (\frac{θ}{2} - \frac{π}{6}) = 1

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