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Popular >

1/(1-sin(x))+1/(1+sin(x))=2csc^2(x)

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 csc^{2} (x)

from both sides

\frac{2 csc ^{2} ( x ) sin ^{2} ( x ) - 2 csc ^{2} ( x ) + 2}{( - sin ( x ) + 1 ) ( sin ( x ) + 1 )} = 0

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

2 - 2 csc^{2} (x) + 2 csc^{2} (x) (\frac{1}{csc ( x )})^{2} : - 2 csc^{2} (x) + 4

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

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Cancel the common factor:

csc^{2} (x)

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

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

Group like terms

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

Add the numbers:

2 + 2 = 4

= - 2 csc^{2} (x) + 4

= - 2 csc^{2} (x) + 4

4 - 2 csc^{2} (x) = 0

Solve by substitution

4 - 2 csc^{2} (x) = 0

Let:

csc (x) = u

4 - 2 u^{2} = 0

4 - 2 u^{2} = 0 : u = 2, u = - 2

4 - 2 u^{2} = 0

Move

4

to the right side

4 - 2 u^{2} = 0

Subtract

4

from both sides

4 - 2 u^{2} - 4 = 0 - 4

Simplify

- 2 u^{2} = - 4

- 2 u^{2} = - 4

Divide both sides by

- 2

- 2 u^{2} = - 4

Divide both sides by

- 2

\frac{- 2 u ^{2}}{- 2} = \frac{- 4}{- 2}

Simplify

\frac{- 2 u ^{2}}{- 2} = \frac{- 4}{- 2}

Simplify

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

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Divide the numbers:

\frac{2}{2} = 1

= u^{2}

Simplify

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

u^{2} = 2

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

Substitute back

u = csc (x)

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = 2

General solutions for

csc (x) = 2

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}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) = - 2

General solutions for

csc (x) = - 2

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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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