What is the general solution for cot(x)-1=csc(x) ?

The general solution for cot(x)-1=csc(x) is x=(3pi)/2+2pin

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cot(x)-1=csc(x)

cot (x) - 1 = csc (x)

x = \frac{3 π}{2} + 2 πn

Solution steps

cot (x) - 1 = csc (x)

Subtract

csc (x)

from both sides

cot (x) - 1 - csc (x) = 0

Express with sin, cos

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

Simplify

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

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

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

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

Add

sin (x)

to both sides

cos (x) - 1 = sin (x)

Square both sides

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

Subtract

sin^{2} (x)

from both sides

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

Factor

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

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

Solving each part separately

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

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

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

Combine all the solutions

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

Verify solutions by plugging them into the original equation

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

Since the equation is undefined for:

2 πn, 2 π + 2 πn

x = \frac{3 π}{2} + 2 πn

- 6 cos^{2} (θ) - 3 cos (θ) + 1 = - 4 cos (θ)

tan (θ) = 1.2

sin (θ) = - \frac{15}{17}

tan (θ) = - \frac{3}{2}

2 cos (2 x) cos (x) + 2 sin (2 x) sin (x) = - 1

What is the general solution for cot(x)-1=csc(x) ?
The general solution for cot(x)-1=csc(x) is x=(3pi)/2+2pin