What is the general solution for tan(x)+tan(x+45)=-2 ?

The general solution for tan(x)+tan(x+45)=-2 is x=120+180n,x=60+180n

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tan(x)+tan(x+45)=-2

tan (x) + tan (x + 4 5^{\circ}) = - 2

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) + tan (x + 4 5^{\circ}) = - 2

Rewrite using trig identities

tan (x) + \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} = - 2

Subtract

- 2

from both sides

tan (x) + \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} + 2 = 0

Simplify

tan (x) + \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} + 2 : \frac{tan ( x ) cos ( x ) - tan ( x ) sin ( x ) - sin ( x ) + 3 cos ( x )}{cos ( x ) - sin ( x )}

\frac{tan ( x ) cos ( x ) - tan ( x ) sin ( x ) - sin ( x ) + 3 cos ( x )}{cos ( x ) - sin ( x )} = 0

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

tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x) = 0

Express with sin, cos

\frac{- sin ^{2} ( x ) + 3 cos ^{2} ( x )}{cos ( x )} = 0

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

- sin^{2} (x) + 3 cos^{2} (x) = 0

Factor

- sin^{2} (x) + 3 cos^{2} (x) : (3 cos (x) + sin (x)) (3 cos (x) - sin (x))

(3 cos (x) + sin (x)) (3 cos (x) - sin (x)) = 0

Solving each part separately

3 cos (x) + sin (x) = 0 or 3 cos (x) - sin (x) = 0

3 cos (x) + sin (x) = 0 : x = 12 0^{\circ} + 18 0^{\circ} n

3 cos (x) - sin (x) = 0 : x = 6 0^{\circ} + 18 0^{\circ} n

Combine all the solutions

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

tan (x) = \frac{22.5}{34}

y, arccot (y) = ln (sin^{2} (3 x)) - e^{x}

tan^{2} (4 x) = 0

cos (\frac{x}{2}) = - 0.4

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

What is the general solution for tan(x)+tan(x+45)=-2 ?
The general solution for tan(x)+tan(x+45)=-2 is x=120+180n,x=60+180n