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

The general solution for (1+tan(x))/(1+cot(x))=2 is x=1.10714…+pin

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(1+tan(x))/(1+cot(x))=2

\frac{1 + tan ( x )}{1 + cot ( x )} = 2

x = 1.10714 \dots + πn

Solution steps

\frac{1 + tan ( x )}{1 + cot ( x )} = 2

Subtract

2

from both sides

\frac{1 + tan ( x )}{1 + cot ( x )} - 2 = 0

Simplify

\frac{1 + tan ( x )}{1 + cot ( x )} - 2 : \frac{tan ( x ) - 2 cot ( x ) - 1}{1 + cot ( x )}

\frac{tan ( x ) - 2 cot ( x ) - 1}{1 + cot ( x )} = 0

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

tan (x) - 2 cot (x) - 1 = 0

Rewrite using trig identities

- 1 + \frac{1}{cot ( x )} - 2 cot (x) = 0

Solve by substitution

cot (x) = - 1, cot (x) = \frac{1}{2}

cot (x) = - 1 : x = \frac{3 π}{4} + πn

cot (x) = \frac{1}{2} : x = arccot (\frac{1}{2}) + πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = arccot (\frac{1}{2}) + πn

Since the equation is undefined for:

\frac{3 π}{4} + πn

x = arccot (\frac{1}{2}) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn

tan (x) + 1 = \frac{1}{3} + \frac{1}{3} cot (x)

sin (8 x) = 1

2 cos^{2} (w) - 3 cos (w) - 5 = 0

4 sin (x) = csc (x)

cot (θ) = - \frac{3}{2}

What is the general solution for (1+tan(x))/(1+cot(x))=2 ?
The general solution for (1+tan(x))/(1+cot(x))=2 is x=1.10714…+pin