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Popular >

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

Solution

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

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Multiply both sides by

1 - u^{2}

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

Multiply both sides by

1 - u^{2}

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

Simplify

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

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

Solve

u^{2} + 1 = 3 (1 - u^{2}) : u = \frac{1}{2}, u = - \frac{1}{2}

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

Expand

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

3 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = u^{2}

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

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 u^{2}

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Move

3 u^{2}

to the left side

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

Add

3 u^{2}

to both sides

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

Simplify

4 u^{2} = 2

4 u^{2} = 2

Divide both sides by

4

4 u^{2} = 2

Divide both sides by

4

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

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

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

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

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

and compare to zero

Solve

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

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

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

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 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 radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

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

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{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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