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

The general solution for (sin^3(x))/(cos^3(x)-cos(x))=tan(x) is No Solution for x\in\mathbb{R}

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(sin^3(x))/(cos^3(x)-cos(x))=tan(x)

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

No Solution for x \in R

Solution steps

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

Subtract

tan (x)

from both sides

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

Simplify

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

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

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

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

Rewrite using trig identities

sin^{3} (x) - (- 1 + cos^{2} (x)) sin (x) = 0

Factor

sin^{3} (x) - (- 1 + cos^{2} (x)) sin (x) : sin (x) (sin^{2} (x) - (cos (x) + 1) (cos (x) - 1))

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) - (cos (x) + 1) (cos (x) - 1) = 0 : x = 2 πn, x = π + 2 πn

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

4 sin (t) - 3 cos (t) = 0

2 - 3 sin (θ) = cos (θ)

5 sin (3 x) = 4

sec (θ) = \frac{1}{cos ( 67.3 8 ^{\circ} )}

x, 6 cos^{2} (x) + cos (x) - 1 = 0

What is the general solution for (sin^3(x))/(cos^3(x)-cos(x))=tan(x) ?
The general solution for (sin^3(x))/(cos^3(x)-cos(x))=tan(x) is No Solution for x\in\mathbb{R}