What is the general solution for tan(x-20)=cot(x+45) ?

The general solution for tan(x-20)=cot(x+45) is x=pin+pi/4-25/2 ,x=pin-25/2+(3pi)/4

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tan(x-20)=cot(x+45)

tan (x - 20) = cot (x + 45)

x = πn + \frac{π}{4} - \frac{25}{2}, x = πn - \frac{25}{2} + \frac{3 π}{4}

Solution steps

tan (x - 20) = cot (x + 45)

Subtract

cot (x + 45)

from both sides

tan (x - 20) - cot (x + 45) = 0

Express with sin, cos

\frac{- cos ( - 20 + x ) cos ( 45 + x ) + sin ( - 20 + x ) sin ( 45 + x )}{cos ( - 20 + x ) sin ( 45 + x )} = 0

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

- cos (- 20 + x) cos (45 + x) + sin (- 20 + x) sin (45 + x) = 0

Rewrite using trig identities

- cos (- 20 + x + 45 + x) = 0

Divide both sides by

- 1

cos (- 20 + x + 45 + x) = 0

General solutions for

cos (- 20 + x + 45 + x) = 0

- 20 + x + 45 + x = \frac{π}{2} + 2 πn, - 20 + x + 45 + x = \frac{3 π}{2} + 2 πn

Solve

- 20 + x + 45 + x = \frac{π}{2} + 2 πn : x = πn + \frac{π}{4} - \frac{25}{2}

Solve

- 20 + x + 45 + x = \frac{3 π}{2} + 2 πn : x = πn - \frac{25}{2} + \frac{3 π}{4}

x = πn + \frac{π}{4} - \frac{25}{2}, x = πn - \frac{25}{2} + \frac{3 π}{4}

sin (z) = 4

- 63 = - 9 csc (3 x)

1 + sin (θ) = 1 + cos (θ)

tan (x) = tan (4 0^{\circ}) tan (5 0^{\circ})

tan (x) + tan (2 x) = 0

What is the general solution for tan(x-20)=cot(x+45) ?
The general solution for tan(x-20)=cot(x+45) is x=pin+pi/4-25/2 ,x=pin-25/2+(3pi)/4