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

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

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Popular >

(1+sin(x))(1+sin(x))=cos^2(x)

Solution

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

Solution

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

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

Factor

(1 + sin (x))^{2} - cos^{2} (x) : (1 + sin (x) + cos (x)) (1 + sin (x) - cos (x))

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

Apply Difference of Two Squares Formula:

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

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

= ((1 + sin (x)) + cos (x)) ((1 + sin (x)) - cos (x))

Refine

= (sin (x) + cos (x) + 1) (sin (x) - cos (x) + 1)

(1 + sin (x) + cos (x)) (1 + sin (x) - cos (x)) = 0

Solving each part separately

1 + sin (x) + cos (x) = 0 or 1 + sin (x) - cos (x) = 0

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

1 + sin (x) + cos (x) = 0

Rewrite using trig identities

1 + sin (x) + cos (x)

sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

sin (x) + cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) + sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= 1 + 2 sin (x + \frac{π}{4})

1 + 2 sin (x + \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (x + \frac{π}{4}) = 0

Subtract

1

from both sides

1 + 2 sin (x + \frac{π}{4}) - 1 = 0 - 1

Simplify

2 sin (x + \frac{π}{4}) = - 1

2 sin (x + \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (x + \frac{π}{4}) = - 1

Divide both sides by

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x + \frac{π}{4})

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

General solutions for

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{5 π}{4} + 2 πn - \frac{π}{4} : 2 πn + π

\frac{5 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{5 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{5 π}{4} : π

Apply rule

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

= \frac{- π + 5 π}{4}

Add similar elements:

- π + 5 π = 4 π

= \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{7 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{7 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{7 π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{- π + 7 π}{4}

Add similar elements:

- π + 7 π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

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

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

1 + sin (x) - cos (x) = 0

Rewrite using trig identities

1 + sin (x) - cos (x)

sin (x) - cos (x) = 2 sin (x - \frac{π}{4})

sin (x) - cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) - sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (x - \frac{π}{4})

= 1 + 2 sin (x - \frac{π}{4})

1 + 2 sin (x - \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (x - \frac{π}{4}) = 0

Subtract

1

from both sides

1 + 2 sin (x - \frac{π}{4}) - 1 = 0 - 1

Simplify

2 sin (x - \frac{π}{4}) = - 1

2 sin (x - \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (x - \frac{π}{4}) = - 1

Divide both sides by

2

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} : sin (x - \frac{π}{4})

\frac{2 sin ( x - \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x - \frac{π}{4})

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x - \frac{π}{4}) = - \frac{2}{2}

sin (x - \frac{π}{4}) = - \frac{2}{2}

sin (x - \frac{π}{4}) = - \frac{2}{2}

General solutions for

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

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

Solve

x - \frac{π}{4} = \frac{5 π}{4} + 2 πn : x = 2 πn + \frac{3 π}{2}

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplify

\frac{5 π}{4} + 2 πn + \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{5 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{5 π}{4}

Combine the fractions

\frac{π}{4} + \frac{5 π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{π + 5 π}{4}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

Solve

x - \frac{π}{4} = \frac{7 π}{4} + 2 πn : x = 2 πn + 2 π

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplify

\frac{7 π}{4} + 2 πn + \frac{π}{4} : 2 πn + 2 π

\frac{7 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

\frac{π}{4} + \frac{7 π}{4} : 2 π

Apply rule

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

= \frac{π + 7 π}{4}

Add similar elements:

π + 7 π = 8 π

= \frac{8 π}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2 π

= 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

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

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

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