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

The general solution for tan(x+20)=cot(x) is x=180n+45-10,x=180n-10+135

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tan(x+20)=cot(x)

tan (x + 2 0^{\circ}) = cot (x)

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}, x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

Solution steps

tan (x + 2 0^{\circ}) = cot (x)

Subtract

cot (x)

from both sides

tan (x + 2 0^{\circ}) - cot (x) = 0

Express with sin, cos

\frac{- cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) cos ( x ) + sin ( \frac{18 0 ^{\circ} + 9 x}{9} ) sin ( x )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) sin ( x )} = 0

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

- cos (\frac{18 0 ^{\circ} + 9 x}{9}) cos (x) + sin (\frac{18 0 ^{\circ} + 9 x}{9}) sin (x) = 0

Rewrite using trig identities

- cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

Divide both sides by

- 1

cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

General solutions for

cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n, \frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

Solve

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

Solve

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}, x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

2 sin (θ) = cot (θ)

3 csc^{2} (x) + 1 = 5

6 cos^{2} (x) - 12 cos (x) + 6 = 0

csc (x) = sin (x) tan (x) + cos (x)

6 sin^{2} (θ) - 10 sin (θ) - 8 = - 3 sin (θ) - 3

What is the general solution for tan(x+20)=cot(x) ?
The general solution for tan(x+20)=cot(x) is x=180n+45-10,x=180n-10+135