What is the general solution for (tan^2(b)+1)/(tan(b))=csc^2(b) ?

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

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(tan^2(b)+1)/(tan(b))=csc^2(b)

Solution

\frac{tan ^{2} ( b ) + 1}{tan ( b )} = csc^{2} (b)

Solution

b = \frac{π}{4} + πn

Degrees

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

Solution steps

\frac{tan ^{2} ( b ) + 1}{tan ( b )} = csc^{2} (b)

Subtract

csc^{2} (b)

from both sides

\frac{tan ^{2} ( b ) + 1}{tan ( b )} - csc^{2} (b) = 0

Simplify

\frac{tan ^{2} ( b ) + 1}{tan ( b )} - csc^{2} (b) : \frac{tan ^{2} ( b ) + 1 - csc ^{2} ( b ) tan ( b )}{tan ( b )}

\frac{tan ^{2} ( b ) + 1}{tan ( b )} - csc^{2} (b)

Convert element to fraction:

csc^{2} (b) = \frac{csc ^{2} ( b ) tan ( b )}{tan ( b )}

= \frac{tan ^{2} ( b ) + 1}{tan ( b )} - \frac{csc ^{2} ( b ) tan ( b )}{tan ( b )}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ^{2} ( b ) + 1 - csc ^{2} ( b ) tan ( b )}{tan ( b )}

\frac{tan ^{2} ( b ) + 1 - csc ^{2} ( b ) tan ( b )}{tan ( b )} = 0

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

tan^{2} (b) + 1 - csc^{2} (b) tan (b) = 0

Rewrite using trig identities

1 + tan^{2} (b) - csc^{2} (b) tan (b)

Use the Pythagorean identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

1 + (\frac{1}{cot ( b )})^{2} - (1 + cot^{2} (b)) \frac{1}{cot ( b )} : 1 + \frac{1}{cot ^{2} ( b )} - \frac{1 + cot ^{2} ( b )}{cot ( b )}

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

(1 + cot^{2} (b)) \frac{1}{cot ( b )}

Multiply fractions:

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

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

1 \cdot (1 + cot^{2} (b)) = 1 + cot^{2} (b)

1 \cdot (1 + cot^{2} (b))

Multiply:

1 \cdot (1 + cot^{2} (b)) = (1 + cot^{2} (b))

= (1 + cot^{2} (b))

Remove parentheses:

(a) = a

= 1 + cot^{2} (b)

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

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

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

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

Solve by substitution

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

Let:

cot (b) = u

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

1 - \frac{1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0 : u = 1, u = i, u = - i

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

Multiply by LCM

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

Find Least Common Multiplier of

u, u^{2} : u^{2}

u, u^{2}

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u

u^{2}

= u^{2}

Multiply by LCM=

u^{2}

1 \cdot u^{2} - \frac{1 + u ^{2}}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

1 \cdot u^{2} - \frac{1 + u ^{2}}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

1 \cdot u^{2} : u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

Simplify

- \frac{1 + u ^{2}}{u} u^{2} : - u (u^{2} + 1)

- \frac{1 + u ^{2}}{u} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u

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

Simplify

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 1

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

u^{2} - u (u^{2} + 1) + 1 = 0 : u = 1, u = i, u = - i

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

Expand

u^{2} - u (u^{2} + 1) + 1 : u^{2} - u^{3} - u + 1

u^{2} - u (u^{2} + 1) + 1

Expand

- u (u^{2} + 1) : - u^{3} - u

- u (u^{2} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - u, b = u^{2}, c = 1

= - u u^{2} + (- u) \cdot 1

Apply minus-plus rules

+ (- a) = - a

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

Simplify

- u^{2} u - 1 \cdot u : - u^{3} - u

- u^{2} u - 1 \cdot u

u^{2} u = u^{3}

u^{2} u

Apply exponent rule:

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

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

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

1 \cdot u = u

1 \cdot u

Multiply:

1 \cdot u = u

= u

= - u^{3} - u

= - u^{3} - u

= u^{2} - u^{3} - u + 1

u^{2} - u^{3} - u + 1 = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + b = 0

- u^{3} + u^{2} - u + 1 = 0

Factor

- u^{3} + u^{2} - u + 1 : - (u - 1) (u^{2} + 1)

- u^{3} + u^{2} - u + 1

Factor out common term

- 1

= - (u^{3} - u^{2} + u - 1)

Factor

u^{3} - u^{2} + u - 1 : (u - 1) (u^{2} + 1)

u^{3} - u^{2} + u - 1

= (u^{3} - u^{2}) + (u - 1)

Factor out

u^{2}

from

u^{3} - u^{2} : u^{2} (u - 1)

u^{3} - u^{2}

Apply exponent rule:

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

u^{3} = u u^{2}

= u u^{2} - u^{2}

Factor out common term

u^{2}

= u^{2} (u - 1)

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

Factor out common term

u - 1

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

Move

1

to the right side

u^{2} + 1 = 0

Subtract

1

from both sides

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

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

The solutions are

u = 1, u = i, u = - i

u = 1, u = i, u = - i

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

1 - \frac{1 + u ^{2}}{u} + \frac{1}{u ^{2}}

and compare to zero

u = 0

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = i, u = - i

Substitute back

u = cot (b)

cot (b) = 1, cot (b) = i, cot (b) = - i

cot (b) = 1, cot (b) = i, cot (b) = - i

cot (b) = 1 : b = \frac{π}{4} + πn

cot (b) = 1

General solutions for

cot (b) = 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

b = \frac{π}{4} + πn

b = \frac{π}{4} + πn

cot (b) = i :

No Solution

cot (b) = i

No Solution

cot (b) = - i :

No Solution

cot (b) = - i

No Solution

Combine all the solutions

b = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

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