What is the general solution for (cos(x))/(1+sin(x))=1-(sin(x))/(cos(x)) ?

The general solution for (cos(x))/(1+sin(x))=1-(sin(x))/(cos(x)) is x=2pin

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

Solution

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

Solution

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

\frac{cos ( x )}{1 + sin ( x )} - 1 + \frac{sin ( x )}{cos ( x )} = 0

Simplify

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

\frac{cos ( x )}{1 + sin ( x )} - 1 + \frac{sin ( x )}{cos ( x )}

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1 + sin (x), 1, cos (x) : cos (x) (sin (x) + 1)

1 + sin (x), 1, cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

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

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos (x) (sin (x) + 1)

For

\frac{cos ( x )}{1 + sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

For

\frac{1}{1} :

multiply the denominator and numerator by

cos (x) (sin (x) + 1)

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

For

\frac{sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x) + 1

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = - cos (x), b = 1, c = sin (x)

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

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot cos (x) - cos (x) sin (x)

Multiply:

1 \cdot cos (x) = cos (x)

= - cos (x) - cos (x) sin (x)

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

Expand

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

sin (x) (1 + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

a = sin (x), b = 1, c = sin (x)

= sin (x) \cdot 1 + sin (x) sin (x)

= 1 \cdot sin (x) + sin (x) sin (x)

Simplify

1 \cdot sin (x) + sin (x) sin (x) : sin (x) + sin^{2} (x)

1 \cdot sin (x) + sin (x) sin (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

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

Factor

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

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

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

Factor out

- sin (x)

from

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

- sin (x) cos (x) + sin (x)

Factor out common term

- sin (x)

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

Factor out

- 1

from

- cos (x) + 1 : - (cos (x) - 1)

- cos (x) + 1

Factor out common term

- 1

= - (cos (x) - 1)

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

Factor out common term

cos (x) - 1

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

Factor

- sin (x) - 1 : - (sin (x) + 1)

- sin (x) - 1

Factor out common term

- 1

= - (sin (x) + 1)

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

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

Solving each part separately

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

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

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

sin (x) + 1 = 0

Move

1

to the right side

sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

sin (x) = - 1

sin (x) = - 1

General solutions for

sin (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = 2 πn, x = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

x = 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (cos(x))/(1+sin(x))=1-(sin(x))/(cos(x)) ?
The general solution for (cos(x))/(1+sin(x))=1-(sin(x))/(cos(x)) is x=2pin