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

The general solution for (1+cot(x))/(1+tan(x))=5 is x=0.19739…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(1+cot(x))/(1+tan(x))=5

\frac{1 + cot ( x )}{1 + tan ( x )} = 5

x = 0.19739 \dots + πn

Solution steps

\frac{1 + cot ( x )}{1 + tan ( x )} = 5

Subtract

5

from both sides

\frac{1 + cot ( x )}{1 + tan ( x )} - 5 = 0

Simplify

\frac{1 + cot ( x )}{1 + tan ( x )} - 5 : \frac{cot ( x ) - 5 tan ( x ) - 4}{1 + tan ( x )}

\frac{cot ( x ) - 5 tan ( x ) - 4}{1 + tan ( x )} = 0

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

cot (x) - 5 tan (x) - 4 = 0

Rewrite using trig identities

- 4 + cot (x) - \frac{5}{cot ( x )} = 0

Solve by substitution

cot (x) = 5, cot (x) = - 1

cot (x) = 5 : x = arccot (5) + πn

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

Combine all the solutions

x = arccot (5) + πn, x = \frac{3 π}{4} + πn

Since the equation is undefined for:

\frac{3 π}{4} + πn

x = arccot (5) + πn

Show solutions in decimal form

x = 0.19739 \dots + πn

4 = 4 cos (x)

cos^{2} (x) = 2 sin (x) - 2

- 3 = 39.6862600000000 \dots cos (x)

1 + cos (θ) = - cos (θ)

2 cos^{2} (x) - cos (- x) - 1 = 0

What is the general solution for (1+cot(x))/(1+tan(x))=5 ?
The general solution for (1+cot(x))/(1+tan(x))=5 is x=0.19739…+pin