What is the general solution for ((1+tan(x)))/(sec(x))=2,x ?

The general solution for ((1+tan(x)))/(sec(x))=2,x is No Solution for x\in\mathbb{R}

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((1+tan(x)))/(sec(x))=2,x

Solution

\frac{( 1 + tan ( x ) )}{sec ( x )} = 2, x

Solution

No Solution for x \in R

Solution steps

\frac{( 1 + tan ( x ) )}{sec ( x )} = 2, x

Subtract

2

from both sides

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

Simplify

\frac{1 + tan ( x )}{sec ( x )} - 2 : \frac{1 + tan ( x ) - 2 sec ( x )}{sec ( x )}

\frac{1 + tan ( x )}{sec ( x )} - 2

Convert element to fraction:

2 = \frac{2 sec ( x )}{sec ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

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

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

1 + tan (x) - 2 sec (x) = 0

Express with sin, cos

1 + tan (x) - 2 sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

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

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

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

2 \cdot \frac{1}{cos ( x )}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

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

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

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

Rewrite using trig identities

- 2 + cos (x) + sin (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})

= - 2 + 2 sin (x + \frac{π}{4})

- 2 + 2 sin (x + \frac{π}{4}) = 0

Move

2

to the right side

- 2 + 2 sin (x + \frac{π}{4}) = 0

Add

2

to both sides

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

Simplify

2 sin (x + \frac{π}{4}) = 2

2 sin (x + \frac{π}{4}) = 2

Divide both sides by

2

2 sin (x + \frac{π}{4}) = 2

Divide both sides by

2

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

Simplify

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

\frac{2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

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

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

Subtract the numbers:

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

= 2^{\frac{1}{2}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2

sin (x + \frac{π}{4}) = 2

sin (x + \frac{π}{4}) = 2

sin (x + \frac{π}{4}) = 2

- 1 \leq sin (x) \leq 1

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for ((1+tan(x)))/(sec(x))=2,x ?
The general solution for ((1+tan(x)))/(sec(x))=2,x is No Solution for x\in\mathbb{R}