What is the general solution for tan((8x+1)/5)*cot((x+7)/2)=1 ?

The general solution for tan((8x+1)/5)*cot((x+7)/2)=1 is x=(3600n)/(11)+3,x=166.63636…+(3600n)/(11)

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tan((8x+1)/5)*cot((x+7)/2)=1

tan (\frac{8 x + 1 ^{\circ}}{5}) \cdot cot (\frac{x + 7 ^{\circ}}{2}) = 1

x = \frac{360 0 ^{\circ} n}{11} + 3^{\circ}, x = 166.63636 \dots^{\circ} + \frac{360 0 ^{\circ} n}{11}

Solution steps

tan (\frac{8 x + 1 ^{\circ}}{5}) cot (\frac{x + 7 ^{\circ}}{2}) = 1

Subtract

1

from both sides

tan (\frac{1440 x + 18 0 ^{\circ}}{900}) cot (\frac{180 x + 126 0 ^{\circ}}{360}) - 1 = 0

Express with sin, cos

\frac{cos ( \frac{180 x + 126 0 ^{\circ}}{360} ) sin ( \frac{18 0 ^{\circ} + 1440 x}{900} ) - cos ( \frac{18 0 ^{\circ} + 1440 x}{900} ) sin ( \frac{180 x + 126 0 ^{\circ}}{360} )}{cos ( \frac{18 0 ^{\circ} + 1440 x}{900} ) sin ( \frac{180 x + 126 0 ^{\circ}}{360} )} = 0

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

cos (\frac{180 x + 126 0 ^{\circ}}{360}) sin (\frac{18 0 ^{\circ} + 1440 x}{900}) - cos (\frac{18 0 ^{\circ} + 1440 x}{900}) sin (\frac{180 x + 126 0 ^{\circ}}{360}) = 0

Rewrite using trig identities

sin (\frac{18 0 ^{\circ} + 1440 x}{900} - \frac{180 x + 126 0 ^{\circ}}{360}) = 0

General solutions for

sin (\frac{18 0 ^{\circ} + 1440 x}{900} - \frac{180 x + 126 0 ^{\circ}}{360}) = 0

\frac{18 0 ^{\circ} + 1440 x}{900} - \frac{180 x + 126 0 ^{\circ}}{360} = 0 + 36 0^{\circ} n, \frac{18 0 ^{\circ} + 1440 x}{900} - \frac{180 x + 126 0 ^{\circ}}{360} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{18 0 ^{\circ} + 1440 x}{900} - \frac{180 x + 126 0 ^{\circ}}{360} = 0 + 36 0^{\circ} n : x = \frac{360 0 ^{\circ} n}{11} + 3^{\circ}

Solve

\frac{18 0 ^{\circ} + 1440 x}{900} - \frac{180 x + 126 0 ^{\circ}}{360} = 18 0^{\circ} + 36 0^{\circ} n : x = 166.63636 \dots^{\circ} + \frac{360 0 ^{\circ} n}{11}

x = \frac{360 0 ^{\circ} n}{11} + 3^{\circ}, x = 166.63636 \dots^{\circ} + \frac{360 0 ^{\circ} n}{11}

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What is the general solution for tan((8x+1)/5)*cot((x+7)/2)=1 ?
The general solution for tan((8x+1)/5)*cot((x+7)/2)=1 is x=(3600n)/(11)+3,x=166.63636…+(3600n)/(11)