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

The general solution for (sin(x)+cos(x))/(sin(x))=1-1/(tan(x)) is No Solution for x\in\mathbb{R}

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

1 - \frac{1}{tan ( x )}

from both sides

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

Simplify

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

sin (x), 1, tan (x) : sin (x) tan (x)

sin (x), 1, tan (x)

Lowest Common Multiplier (LCM)

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

= sin (x) tan (x)

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 (x) tan (x)

For

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

multiply the denominator and numerator by

tan (x)

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

For

\frac{1}{1} :

multiply the denominator and numerator by

sin (x) tan (x)

\frac{1}{1} = \frac{1 \cdot sin ( x ) tan ( x )}{1 \cdot sin ( x ) tan ( x )} = \frac{sin ( x ) tan ( x )}{sin ( x ) tan ( x )}

For

\frac{1}{tan ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{1}{tan ( x )} = \frac{1 \cdot sin ( x )}{tan ( x ) sin ( x )} = \frac{sin ( x )}{sin ( x ) tan ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

Expand

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

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

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

Expand

tan (x) (sin (x) + cos (x)) : tan (x) sin (x) + tan (x) cos (x)

tan (x) (sin (x) + cos (x))

Apply the distributive law:

a (b + c) = ab + a c

a = tan (x), b = sin (x), c = cos (x)

= tan (x) sin (x) + tan (x) cos (x)

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

Add similar elements:

tan (x) sin (x) - sin (x) tan (x) = 0

= tan (x) cos (x) + sin (x)

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

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

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

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

Rewrite using trig identities

sin (x) + cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= sin (x) + sin (x)

Add similar elements:

sin (x) + sin (x) = 2 sin (x)

= 2 sin (x)

= 2 sin (x)

2 sin (x) = 0

Divide both sides by

2

2 sin (x) = 0

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{0}{2}

Simplify

sin (x) = 0

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(x)+cos(x))/(sin(x))=1-1/(tan(x)) ?
The general solution for (sin(x)+cos(x))/(sin(x))=1-1/(tan(x)) is No Solution for x\in\mathbb{R}