What is the general solution for (1+csc(y))/(cot(y)+cos(y))=csc(y) ?

The general solution for (1+csc(y))/(cot(y)+cos(y))=csc(y) is y= pi/4+pin

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(1+csc(y))/(cot(y)+cos(y))=csc(y)

Solution

\frac{1 + csc ( y )}{cot ( y ) + cos ( y )} = csc (y)

Solution

y = \frac{π}{4} + πn

Degrees

y = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{1 + csc ( y )}{cot ( y ) + cos ( y )} = csc (y)

Subtract

csc (y)

from both sides

\frac{1 + csc ( y )}{cot ( y ) + cos ( y )} - csc (y) = 0

Simplify

\frac{1 + csc ( y )}{cot ( y ) + cos ( y )} - csc (y) : \frac{1 + csc ( y ) - csc ( y ) ( cot ( y ) + cos ( y ) )}{cot ( y ) + cos ( y )}

\frac{1 + csc ( y )}{cot ( y ) + cos ( y )} - csc (y)

Convert element to fraction:

csc (y) = \frac{csc ( y ) ( cot ( y ) + cos ( y ) )}{cot ( y ) + cos ( y )}

= \frac{1 + csc ( y )}{cot ( y ) + cos ( y )} - \frac{csc ( y ) ( cot ( y ) + cos ( y ) )}{cot ( y ) + cos ( y )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + csc ( y ) - csc ( y ) ( cot ( y ) + cos ( y ) )}{cot ( y ) + cos ( y )}

\frac{1 + csc ( y ) - csc ( y ) ( cot ( y ) + cos ( y ) )}{cot ( y ) + cos ( y )} = 0

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

1 + csc (y) - csc (y) (cot (y) + cos (y)) = 0

Express with sin, cos

1 + csc (y) - (cos (y) + cot (y)) csc (y)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

Simplify

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

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

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

(cos (y) + \frac{cos ( y )}{sin ( y )}) \frac{1}{sin ( y )}

Multiply fractions:

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

= \frac{1 \cdot ( cos ( y ) + \frac{c o s ( y )}{s i n ( y )} )}{sin ( y )}

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

1 \cdot (cos (y) + \frac{cos ( y )}{sin ( y )})

Multiply:

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

= (cos (y) + \frac{cos ( y )}{sin ( y )})

Remove parentheses:

(a) = a

= cos (y) + \frac{cos ( y )}{sin ( y )}

= \frac{cos ( y ) + \frac{c o s ( y )}{s i n ( y )}}{sin ( y )}

Join

cos (y) + \frac{cos ( y )}{sin ( y )} : \frac{cos ( y ) sin ( y ) + cos ( y )}{sin ( y )}

cos (y) + \frac{cos ( y )}{sin ( y )}

Convert element to fraction:

cos (y) = \frac{cos ( y ) sin ( y )}{sin ( y )}

= \frac{cos ( y ) sin ( y )}{sin ( y )} + \frac{cos ( y )}{sin ( y )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( y ) sin ( y ) + cos ( y )}{sin ( y )}

= \frac{\frac{c o s ( y ) s i n ( y ) + c o s ( y )}{s i n ( y )}}{sin ( y )}

Apply the fraction rule:

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

= \frac{cos ( y ) sin ( y ) + cos ( y )}{sin ( y ) sin ( y )}

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

sin (y) sin (y)

Apply exponent rule:

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

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

= sin^{1 + 1} (y)

Add the numbers:

1 + 1 = 2

= sin^{2} (y)

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, sin (y), sin^{2} (y) : sin^{2} (y)

1, sin (y), sin^{2} (y)

Lowest Common Multiplier (LCM)

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

= sin^{2} (y)

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

sin^{2} (y)

For

\frac{1}{1} :

multiply the denominator and numerator by

sin^{2} (y)

\frac{1}{1} = \frac{1 \cdot sin ^{2} ( y )}{1 \cdot sin ^{2} ( y )} = \frac{sin ^{2} ( y )}{sin ^{2} ( y )}

For

\frac{1}{sin ( y )} :

multiply the denominator and numerator by

sin (y)

\frac{1}{sin ( y )} = \frac{1 \cdot sin ( y )}{sin ( y ) sin ( y )} = \frac{sin ( y )}{sin ^{2} ( y )}

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

Since the denominators are equal, combine the fractions:

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

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

- (cos (y) sin (y) + cos (y)) : - cos (y) sin (y) - cos (y)

- (cos (y) sin (y) + cos (y))

Distribute parentheses

= - (cos (y) sin (y)) - (cos (y))

Apply minus-plus rules

+ (- a) = - a

= - cos (y) sin (y) - cos (y)

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

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

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

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

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

Factor

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

- cos (y) + sin (y) + sin^{2} (y) - cos (y) sin (y)

Factor out common term

cos (y)

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

Apply exponent rule:

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

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

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

Factor out common term

sin (y)

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

Factor out common term

(1 + sin (y))

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

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

Solving each part separately

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

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

1 + sin (y) = 0

Move

1

to the right side

1 + sin (y) = 0

Subtract

1

from both sides

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

Simplify

sin (y) = - 1

sin (y) = - 1

General solutions for

sin (y) = - 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}

y = \frac{3 π}{2} + 2 πn

y = \frac{3 π}{2} + 2 πn

- cos (y) + sin (y) = 0 : y = \frac{π}{4} + πn

- cos (y) + sin (y) = 0

Rewrite using trig identities

- cos (y) + sin (y) = 0

Divide both sides by

cos (y), cos (y) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 1 + tan (y) = 0

- 1 + tan (y) = 0

Move

1

to the right side

- 1 + tan (y) = 0

Add

1

to both sides

- 1 + tan (y) + 1 = 0 + 1

Simplify

tan (y) = 1

tan (y) = 1

General solutions for

tan (y) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

y = \frac{π}{4} + πn

y = \frac{π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

y = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (1+csc(y))/(cot(y)+cos(y))=csc(y) ?
The general solution for (1+csc(y))/(cot(y)+cos(y))=csc(y) is y= pi/4+pin