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

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

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sec^2(x)-1= 1/(cot(x))

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

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

Solution steps

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

Square both sides

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

Subtract

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

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

sec^{2} (x) = tan^{2} (x) + 1

sec^{2} (x) - 1 = tan^{2} (x)

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

(tan^{2} (x))^{2} = tan^{4} (x)

(tan^{2} (x))^{2}

Apply exponent rule:

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

= tan^{2 \cdot 2} (x)

Multiply the numbers:

2 \cdot 2 = 4

= tan^{4} (x)

= - 1 + tan^{4} (x) cot^{2} (x)

- 1 + cot^{2} (x) tan^{4} (x) = 0

Factor

- 1 + cot^{2} (x) tan^{4} (x) : (tan^{2} (x) cot (x) + 1) (tan^{2} (x) cot (x) - 1)

- 1 + cot^{2} (x) tan^{4} (x)

Rewrite

- 1 + cot^{2} (x) tan^{4} (x)

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

- 1 + cot^{2} (x) tan^{4} (x)

Apply exponent rule:

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

tan^{4} (x) = (tan^{2} (x))^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

Solving each part separately

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

tan^{2} (x) cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

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

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

tan^{2} (x) cot (x) - 1 = 0 : x = \frac{π}{4} + πn

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

Combine all the solutions

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

Remove the ones that don't agree with the equation.

Check the solution

\frac{3 π}{4} + πn :

False

\frac{3 π}{4} + πn

Plug in

n = 1

\frac{3 π}{4} + π 1

For

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

plug in

x = \frac{3 π}{4} + π 1

sec^{2} (\frac{3 π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{3 π}{4} + π 1 )}

Refine

1 = - 1

\Rightarrow False

Check the solution

\frac{π}{4} + πn :

True

\frac{π}{4} + πn

Plug in

n = 1

\frac{π}{4} + π 1

For

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

plug in

x = \frac{π}{4} + π 1

sec^{2} (\frac{π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{π}{4} + π 1 )}

Refine

1 = 1

\Rightarrow True

x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

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