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

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

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

- (sin (x) + 1)

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - sin (x) - 1

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

- 1 - sin (x) + cos (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)

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

Group like terms

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

Add similar elements:

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

= cos (x) sin (x) - 1

= cos (x) sin (x) - 1

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

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