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Popular >

1/(csc^2(a))=1-sin^2(a)

Solution

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

Solution

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

Degrees

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

Solution steps

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

Subtract

1 - sin^{2} (a)

from both sides

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

Simplify

\frac{1}{csc ^{2} ( a )} - 1 + sin^{2} (a) : \frac{1 - csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

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

Convert element to fraction:

1 = \frac{1 csc ^{2} ( a )}{csc ^{2} ( a )}, sin^{2} (a) = \frac{sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot csc^{2} (a) = csc^{2} (a)

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

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

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

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

Rewrite using trig identities

1 - csc^{2} (a) + csc^{2} (a) sin^{2} (a)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Cancel the common factor:

csc^{2} (a)

= 1

= 1 - csc^{2} (a) + 1

Group like terms

= - csc^{2} (a) + 1 + 1

Add the numbers:

1 + 1 = 2

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

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

2 - csc^{2} (a) = 0

Solve by substitution

2 - csc^{2} (a) = 0

Let:

csc (a) = u

2 - u^{2} = 0

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

2 - u^{2} = 0

Move

2

to the right side

2 - u^{2} = 0

Subtract

2

from both sides

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

Simplify

- u^{2} = - 2

- u^{2} = - 2

Divide both sides by

- 1

- u^{2} = - 2

Divide both sides by

- 1

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

Simplify

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 (a)

csc (a) = 2, csc (a) = - 2

csc (a) = 2, csc (a) = - 2

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

csc (a) = 2

General solutions for

csc (a) = 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

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

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

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

csc (a) = - 2

General solutions for

csc (a) = - 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

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

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

Combine all the solutions

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

Graph

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