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

The general solution for 1/((1+tan^2(o)+1))+1/((1+cot^2(o)))=1 is No Solution for o\in\mathbb{R}

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1/((1+tan^2(o)+1))+1/((1+cot^2(o)))=1

Solution

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

Solution

No Solution for o \in R

Solution steps

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

Subtract

1

from both sides

\frac{cot ^{2} ( o ) + tan ^{2} ( o ) + 3}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )} - 1 = 0

Simplify

\frac{cot ^{2} ( o ) + tan ^{2} ( o ) + 3}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )} - 1 : \frac{- cot ^{2} ( o ) - tan ^{2} ( o ) cot ^{2} ( o ) + 1}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}

\frac{cot ^{2} ( o ) + tan ^{2} ( o ) + 3}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )} - 1

Convert element to fraction:

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

= \frac{cot ^{2} ( o ) + tan ^{2} ( o ) + 3}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )} - \frac{1 \cdot ( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}

Since the denominators are equal, combine the fractions:

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

= \frac{cot ^{2} ( o ) + tan ^{2} ( o ) + 3 - 1 \cdot ( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}

Multiply:

1 \cdot (tan^{2} (o) + 2) = (tan^{2} (o) + 2)

= \frac{cot ^{2} ( o ) + tan ^{2} ( o ) + 3 - ( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}{( tan ^{2} ( o ) + 2 ) ( cot ^{2} ( o ) + 1 )}

Expand

cot^{2} (o) + tan^{2} (o) + 3 - (tan^{2} (o) + 2) (cot^{2} (o) + 1) : - cot^{2} (o) - tan^{2} (o) cot^{2} (o) + 1

cot^{2} (o) + tan^{2} (o) + 3 - (tan^{2} (o) + 2) (cot^{2} (o) + 1)

Expand

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

Expand

(tan^{2} (o) + 2) (cot^{2} (o) + 1) : tan^{2} (o) cot^{2} (o) + tan^{2} (o) + 2 cot^{2} (o) + 2

(tan^{2} (o) + 2) (cot^{2} (o) + 1)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = tan^{2} (o), b = 2, c = cot^{2} (o), d = 1

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

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

Simplify

tan^{2} (o) cot^{2} (o) + 1 \cdot tan^{2} (o) + 2 cot^{2} (o) + 2 \cdot 1 : tan^{2} (o) cot^{2} (o) + tan^{2} (o) + 2 cot^{2} (o) + 2

tan^{2} (o) cot^{2} (o) + 1 \cdot tan^{2} (o) + 2 cot^{2} (o) + 2 \cdot 1

Multiply:

1 \cdot tan^{2} (o) = tan^{2} (o)

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

Multiply the numbers:

2 \cdot 1 = 2

= tan^{2} (o) cot^{2} (o) + tan^{2} (o) + 2 cot^{2} (o) + 2

= tan^{2} (o) cot^{2} (o) + tan^{2} (o) + 2 cot^{2} (o) + 2

= - (tan^{2} (o) cot^{2} (o) + tan^{2} (o) + 2 cot^{2} (o) + 2)

Distribute parentheses

= - (tan^{2} (o) cot^{2} (o)) - (tan^{2} (o)) - (2 cot^{2} (o)) - (2)

Apply minus-plus rules

+ (- a) = - a

= - tan^{2} (o) cot^{2} (o) - tan^{2} (o) - 2 cot^{2} (o) - 2

= cot^{2} (o) + tan^{2} (o) + 3 - tan^{2} (o) cot^{2} (o) - tan^{2} (o) - 2 cot^{2} (o) - 2

Simplify

cot^{2} (o) + tan^{2} (o) + 3 - tan^{2} (o) cot^{2} (o) - tan^{2} (o) - 2 cot^{2} (o) - 2 : - cot^{2} (o) - tan^{2} (o) cot^{2} (o) + 1

cot^{2} (o) + tan^{2} (o) + 3 - tan^{2} (o) cot^{2} (o) - tan^{2} (o) - 2 cot^{2} (o) - 2

Group like terms

= cot^{2} (o) + tan^{2} (o) - tan^{2} (o) cot^{2} (o) - tan^{2} (o) - 2 cot^{2} (o) + 3 - 2

Add similar elements:

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

= - cot^{2} (o) + tan^{2} (o) - tan^{2} (o) cot^{2} (o) - tan^{2} (o) + 3 - 2

Add similar elements:

tan^{2} (o) - tan^{2} (o) = 0

= - cot^{2} (o) - tan^{2} (o) cot^{2} (o) + 3 - 2

Add/Subtract the numbers:

3 - 2 = 1

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

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

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

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

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

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

Rewrite using trig identities

1 - cot^{2} (o) - cot^{2} (o) tan^{2} (o)

Use the basic trigonometric identity:

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

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

Simplify

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

1 - cot^{2} (o) - cot^{2} (o) (\frac{1}{cot ( o )})^{2}

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Cancel the common factor:

cot^{2} (o)

= 1

= 1 - cot^{2} (o) - 1

Group like terms

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

1 - 1 = 0

= - cot^{2} (o)

= - cot^{2} (o)

- cot^{2} (o) = 0

Divide both sides by

- 1

- cot^{2} (o) = 0

Divide both sides by

- 1

- cot^{2} (o) = 0

Divide both sides by

- 1

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

Simplify

cot^{2} (o) = 0

cot^{2} (o) = 0

Apply rule

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

cot (o) = 0

General solutions for

cot (o) = 0

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

o = \frac{π}{2} + πn

o = \frac{π}{2} + πn

Since the equation is undefined for:

\frac{π}{2} + πn

No Solution for o \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 1/((1+tan^2(o)+1))+1/((1+cot^2(o)))=1 ?
The general solution for 1/((1+tan^2(o)+1))+1/((1+cot^2(o)))=1 is No Solution for o\in\mathbb{R}