What is the general solution for cot(x)=cot(3x-50),0<x<180 ?

The general solution for cot(x)=cot(3x-50),0<x<180 is x=25,x=115

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cot(x)=cot(3x-50),0<x<180

cot (x) = cot (3 x - 5 0^{\circ}), 0 < x < 18 0^{\circ}

x = 2 5^{\circ}, x = 11 5^{\circ}

Solution steps

cot (x) = cot (3 x - 5 0^{\circ}), 0 < x < 18 0^{\circ}

Subtract

cot (3 x - 5 0^{\circ})

from both sides

cot (x) - cot (3 x - 5 0^{\circ}) = 0

Express with sin, cos

\frac{- cos ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x ) + cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x )} = 0

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

- cos (\frac{54 x - 90 0 ^{\circ}}{18}) sin (x) + cos (x) sin (\frac{54 x - 90 0 ^{\circ}}{18}) = 0

Rewrite using trig identities

sin (\frac{54 x - 90 0 ^{\circ}}{18} - x) = 0

General solutions for

sin (\frac{54 x - 90 0 ^{\circ}}{18} - x) = 0

\frac{54 x - 90 0 ^{\circ}}{18} - x = 0 + 36 0^{\circ} n, \frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{54 x - 90 0 ^{\circ}}{18} - x = 0 + 36 0^{\circ} n : x = 18 0^{\circ} n + 2 5^{\circ}

Solve

\frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n : x = 11 5^{\circ} + 18 0^{\circ} n

x = 18 0^{\circ} n + 2 5^{\circ}, x = 11 5^{\circ} + 18 0^{\circ} n

Solutions for the range

0 < x < 18 0^{\circ}

x = 2 5^{\circ}, x = 11 5^{\circ}

tan (2 θ) = \frac{4}{3}

3 sec^{2} (x) + 2 - 8 tan (x) = 0

cos (θ) - 1 = cos (2 θ)

5 tan (x) = 9 sin (x)

x, sin (\frac{π x}{4}) = y - 1

What is the general solution for cot(x)=cot(3x-50),0<x<180 ?
The general solution for cot(x)=cot(3x-50),0<x<180 is x=25,x=115