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

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

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

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

Solution steps

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

Subtract

\frac{1}{cot ( x )}

from both sides

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

Simplify

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

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

Convert element to fraction:

tan^{2} (x) = \frac{tan ^{2} ( x ) cot ( x )}{cot ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

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

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

tan^{2} (x) cot (x) - 1 = 0

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

cot (x) (\frac{1}{cot ( x )})^{2} = \frac{1}{cot ( x )}

cot (x) (\frac{1}{cot ( x )})^{2}

(\frac{1}{cot ( x )})^{2} = \frac{1}{cot ^{2} ( x )}

(\frac{1}{cot ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cot ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( x )}

= \frac{1}{cot ^{2} ( x )} cot (x)

Multiply fractions:

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

= \frac{1 \cdot cot ( x )}{cot ^{2} ( x )}

Multiply:

1 \cdot cot (x) = cot (x)

= \frac{cot ( x )}{cot ^{2} ( x )}

Cancel the common factor:

cot (x)

= \frac{1}{cot ( x )}

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

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

Multiply both sides by

cot (x)

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

Multiply both sides by

cot (x)

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

Simplify

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

Simplify

- 1 \cdot cot (x) : - cot (x)

- 1 \cdot cot (x)

Multiply:

1 \cdot cot (x) = cot (x)

= - cot (x)

Simplify

\frac{1}{cot ( x )} cot (x) : 1

\frac{1}{cot ( x )} cot (x)

Multiply fractions:

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

= \frac{1 \cdot cot ( x )}{cot ( x )}

Cancel the common factor:

cot (x)

= 1

Simplify

0 \cdot cot (x) : 0

0 \cdot cot (x)

Apply rule

0 \cdot a = 0

= 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

Move

1

to the right side

- cot (x) + 1 = 0

Subtract

1

from both sides

- cot (x) + 1 - 1 = 0 - 1

Simplify

- cot (x) = - 1

- cot (x) = - 1

Divide both sides by

- 1

- cot (x) = - 1

Divide both sides by

- 1

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

Simplify

cot (x) = 1

cot (x) = 1

Verify Solutions

Find undefined (singularity) points:

cot (x) = 0

Take the denominator(s) of

- 1 + \frac{1}{cot ( x )}

and compare to zero

cot (x) = 0

The following points are undefined

cot (x) = 0

Combine undefined points with solutions:

cot (x) = 1

General solutions for

cot (x) = 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Popular Examples

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Frequently Asked Questions (FAQ)

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