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

The general solution for tan(x)+1/(tan(x))=2 is x= pi/4+pin

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

u + \frac{1}{u} = 2

u + \frac{1}{u} = 2 : u = 1

u + \frac{1}{u} = 2

Multiply both sides by

u

u + \frac{1}{u} = 2

Multiply both sides by

u

uu + \frac{1}{u} u = 2 u

Simplify

uu + \frac{1}{u} u = 2 u

Simplify

uu : u^{2}

uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

u^{2} + 1 = 2 u

u^{2} + 1 = 2 u

u^{2} + 1 = 2 u

Solve

u^{2} + 1 = 2 u : u = 1

u^{2} + 1 = 2 u

Move

2 u

to the left side

u^{2} + 1 = 2 u

Subtract

2 u

from both sides

u^{2} + 1 - 2 u = 2 u - 2 u

Simplify

u^{2} + 1 - 2 u = 0

u^{2} + 1 - 2 u = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} - 2 u + 1 = 0

Solve with the quadratic formula

u^{2} - 2 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 2, c = 1

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 2)^{2} - 4 \cdot 1 \cdot 1 = 0

(- 2)^{2} - 4 \cdot 1 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

= 2^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} - 4

2^{2} = 4

= 4 - 4

Subtract the numbers:

4 - 4 = 0

= 0

u_{1, 2} = \frac{- ( - 2 ) \pm 0}{2 \cdot 1}

u = \frac{- ( - 2 )}{2 \cdot 1}

\frac{- ( - 2 )}{2 \cdot 1} = 1

\frac{- ( - 2 )}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = 1

The solution to the quadratic equation is:

u = 1

u = 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u + \frac{1}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1

Substitute back

u = tan (x)

tan (x) = 1

tan (x) = 1

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn

Popular Examples

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4 csc (θ) - 5 = 3

csc(5x)=-1,0<= x<= 2pi

csc (5 x) = - 1, 0 \leq x \leq 2 π

-4*cos^2(x)+3=0

- 4 \cdot cos^{2} (x) + 3 = 0

8sin(θ)+1=6sin(θ)

8 sin (θ) + 1 = 6 sin (θ)

-cot^2(x)=1+2cot(x)

- cot^{2} (x) = 1 + 2 cot (x)

Frequently Asked Questions (FAQ)

What is the general solution for tan(x)+1/(tan(x))=2 ?
The general solution for tan(x)+1/(tan(x))=2 is x= pi/4+pin