What is the general solution for cot(x)-sqrt(3)=csc(x) ?

The general solution for cot(x)-sqrt(3)=csc(x) is x=(4pi)/3+2pin

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cot(x)-sqrt(3)=csc(x)

cot (x) - 3 = csc (x)

x = \frac{4 π}{3} + 2 πn

Solution steps

cot (x) - 3 = csc (x)

Subtract

csc (x)

from both sides

cot (x) - 3 - csc (x) = 0

Express with sin, cos

\frac{cos ( x )}{sin ( x )} - 3 - \frac{1}{sin ( x )} = 0

Simplify

\frac{cos ( x )}{sin ( x )} - 3 - \frac{1}{sin ( x )} : \frac{cos ( x ) - 1 - 3 sin ( x )}{sin ( x )}

\frac{cos ( x ) - 1 - 3 sin ( x )}{sin ( x )} = 0

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

cos (x) - 1 - 3 sin (x) = 0

Add

3 sin (x)

to both sides

cos (x) - 1 = 3 sin (x)

Square both sides

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

Subtract

(3 sin (x))^{2}

from both sides

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = 1, cos (x) = - \frac{1}{2}

cos (x) = 1 : x = 2 πn

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

x = 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Verify solutions by plugging them into the original equation

x = 2 πn, x = \frac{4 π}{3} + 2 πn

Since the equation is undefined for:

2 πn

x = \frac{4 π}{3} + 2 πn

cos (b) = 0.47

- 6 = - 5 + cos (θ)

tan (x) = - 26, \frac{3 π}{2} < x < 2 π

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

tan^{2} (x) - 4 sec (x) = 4

What is the general solution for cot(x)-sqrt(3)=csc(x) ?
The general solution for cot(x)-sqrt(3)=csc(x) is x=(4pi)/3+2pin