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

The general solution for 1/1+cot^2(x)=sin^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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1/1+cot^2(x)=sin^2(x)

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

sin^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= - sin^{2} (x) + csc^{2} (x)

csc^{2} (x) - sin^{2} (x) = 0

Factor

csc^{2} (x) - sin^{2} (x) : (csc (x) + sin (x)) (csc (x) - sin (x))

csc^{2} (x) - sin^{2} (x)

Apply Difference of Two Squares Formula:

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

csc^{2} (x) - sin^{2} (x) = (csc (x) + sin (x)) (csc (x) - sin (x))

= (csc (x) + sin (x)) (csc (x) - sin (x))

(csc (x) + sin (x)) (csc (x) - sin (x)) = 0

Solving each part separately

csc (x) + sin (x) = 0 or csc (x) - sin (x) = 0

csc (x) + sin (x) = 0 :

No Solution

csc (x) + sin (x) = 0

Rewrite using trig identities

csc (x) + sin (x)

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

= csc (x) + \frac{1}{csc ( x )}

csc (x) + \frac{1}{csc ( x )} = 0

Solve by substitution

csc (x) + \frac{1}{csc ( x )} = 0

Let:

csc (x) = u

u + \frac{1}{u} = 0

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

u + \frac{1}{u} = 0

Multiply both sides by

u

u + \frac{1}{u} = 0

Multiply both sides by

u

uu + \frac{1}{u} u = 0 \cdot u

Simplify

uu + \frac{1}{u} u = 0 \cdot 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

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} + 1 = 0

u^{2} + 1 = 0

u^{2} + 1 = 0

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

u = i, u = - i

Substitute back

u = csc (x)

csc (x) = i, csc (x) = - i

csc (x) = i, csc (x) = - i

csc (x) = i :

No Solution

csc (x) = i

No Solution

csc (x) = - i :

No Solution

csc (x) = - i

No Solution

Combine all the solutions

No Solution

csc (x) - sin (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

csc (x) - sin (x) = 0

Rewrite using trig identities

csc (x) - sin (x)

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

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

csc (x) - \frac{1}{csc ( x )} = 0

Solve by substitution

csc (x) - \frac{1}{csc ( x )} = 0

Let:

csc (x) = u

u - \frac{1}{u} = 0

u - \frac{1}{u} = 0 : u = 1, u = - 1

u - \frac{1}{u} = 0

Multiply both sides by

u

u - \frac{1}{u} = 0

Multiply both sides by

u

uu - \frac{1}{u} u = 0 \cdot u

Simplify

uu - \frac{1}{u} u = 0 \cdot 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

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 1 = 0

u^{2} - 1 = 0

u^{2} - 1 = 0

Solve

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

u^{2} - 1 = 0

Move

1

to the right side

u^{2} - 1 = 0

Add

1

to 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

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

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, u = - 1

Substitute back

u = csc (x)

csc (x) = 1, csc (x) = - 1

csc (x) = 1, csc (x) = - 1

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

General solutions for

csc (x) = 1

csc (x)

periodicity table with

2 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

General solutions for

csc (x) = - 1

csc (x)

periodicity table with

2 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 1/1+cot^2(x)=sin^2(x) ?
The general solution for 1/1+cot^2(x)=sin^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin